7.4 Confidence Interval (Home Costs)

Aligned outcomes:

Learning Objectives

By the end of this section, you will be able to:

In this section, you will learn to:

collect a real sample of home sale prices from your own area and record it in an organized table;
calculate a 90% confidence interval for the mean cost of a home in that area, together with its error bound;
interpret a confidence interval in plain language, both in general and for this particular study;
determine what happens to the error bound and the width of the interval as the confidence level changes.

7.4.1 Stats Lab: Confidence Interval for Home Costs

This section is a lab, not a reading. Sections 8.1 through 8.3 handed you a sample mean and a standard deviation already computed and asked you to build an interval around them. Here you go get the sample yourself, and then run the whole chapter over it: the sampling logic from §7.1, the Student t machinery from §7.2 because you will not know the population standard deviation, and the interpretation language the whole chapter has been building toward.

Class Time:

Names:

The question is one nobody in your class can answer off the top of their head: what does a house cost around here? People have a rough feeling for it, and that feeling is exactly the thing you are about to replace with a number and an honest margin of error.

You will never see the parameter

The true mean sale price of every home in the county is a fixed number that exists right now, and no amount of sampling will ever show it to you. The interval you build this hour is not a guess at that number. It is a range produced by a procedure that lands on the truth 90% of the time.

That distinction is the whole point of the lab, and it is worth holding onto before the arithmetic starts, because the arithmetic is easy and the interpretation is where almost everyone slips. You are going to compute one interval from one sample. That interval either contains the true mean or it does not — there is no probability left in it once the numbers are on the page. What the 90% describes is the method: if every group in your class collects 35 prices and builds an interval the same way, about nine out of every ten of those intervals will contain the true mean, and none of you will know which ones. The confidence lives in the procedure that generated the interval, not in the particular pair of numbers you end up writing down. Keep that sentence nearby; §7.4.5 will ask you to write your own version of it.

Every step of this lab is easier to do on your own data once you have watched it done on somebody else's. The 35 prices below come from a class that ran this lab in Chico before you, already sorted from smallest to largest. Each Try It Now that follows works one stage of the lab on these numbers, so keep the table where you can see it.

**Table 7.4.1** — Sale prices of 35 randomly selected homes recently listed in Butte County, sorted from smallest to largest.
Position	Sale prices
1-7	$259,000, $272,000, $285,000, $299,000, $305,000, $312,000, $319,000
8-14	$325,000, $330,000, $338,000, $345,000, $349,000, $355,000, $362,000
15-21	$368,000, $375,000, $379,000, $385,000, $392,000, $399,000, $405,000
22-28	$412,000, $420,000, $429,000, $438,000, $449,000, $462,000, $475,000
29-35	$489,000, $505,000, $525,000, $549,000, $585,000, $659,000, $795,000

Try It Now 7.4.1

Bee Vang is starting their write-up for this lab and needs the four pieces named before they compute anything. For the demonstration data in Table 7.4.1, name the population, the sample, the parameter they are after, and the statistic they will use to estimate it. Then say why the parameter cannot simply be looked up.

Solution — naming the four pieces of the study

Step 1 — the population. Every home recently listed for sale in Butte County. That is the whole collection the question is about.

Step 2 — the sample. The 35 prices in Table 7.4.1 — the subset actually examined.

Step 3 — the parameter. $\mu$, the true mean sale price over that entire population of listings. It is a single fixed number.

Step 4 — the statistic. $\overline{x}$, the mean of the 35 prices, which Bee will compute in §7.4.3. It is the estimate that stands in for $\mu$.

Answer: population = all recently listed Butte County homes, sample = these 35 prices, parameter = $\mu$, statistic = $\overline{x}$. The parameter cannot be looked up because computing it would mean recording the price of every listing in the county on the same day and averaging them — the census that sampling exists to avoid. The gap between the $\overline{x}$ they can compute and the $\mu$ they cannot is exactly the gap the confidence interval is built to measure.

7.4.2 Collect the Data

Check the Real Estate section in your local newspaper. Record the sale prices for 35 randomly selected homes recently listed in the county.

The listings are not a random sample of houses

Newspapers print listings on one day a week, and only homes whose sellers chose to advertise there appear at all. We will treat them as random because the lab needs a starting point, but that assumption is a real one.

Write the assumption down rather than skipping past it, because it changes what your final sentence is allowed to claim. A newspaper real-estate section over-represents whatever kinds of homes the local agencies advertise that way and under-represents the rest — a for-sale-by-owner listing on a website, a home sold before it was ever advertised, a rural property listed in a different county's paper. None of that makes the exercise pointless. It makes the population narrower than "homes in the county": what you are really sampling is homes recently listed in this newspaper, and that is the population your interval describes. Statisticians do this constantly. The honest move is not to pretend the sample is broader than it is; it is to name the population your sampling method actually reaches, and to let the reader decide how far to carry the result. If your class wants a genuinely random sample of county listings, a public multiple-listing database with a random-number generator would get you closer, and that is worth a sentence in your write-up either way.

Try it in rāSHio

Deciding which 35 listings to record is the part of this lab most likely to go wrong by hand. Number the listings in your paper, then in rāSHio choose File → Random Numbers…, set the range to run from 1 to however many listings the section carries, ask for 35 of them with No repeats checked, and click Generate — that draw is what earns you the word “random” in every sentence you write afterwards.

Figure 7.4.1 — Drawing which listings to record in rāSHio: File → Random Numbers… The walkthrough draws its own small sample; the steps are the ones you run on your paper’s listing count.

1. Complete the table.

**Table 7.4.2** — Your own sample: the sale prices of 35 randomly selected homes recently listed in your county.
Home sale prices — 35 listings

Try it in rāSHio

Get the 35 prices off your data sheet and into rāSHio once, then reuse that column for every step below: choose File → Delimited List… and type or paste them separated by commas, spaces, or one per line. Enter them as plain numbers — 259000, not $259,000 — because a dollar sign or a comma makes the column text, and every statistic in the rest of the lab then comes back blank.

Figure 7.4.2 — Getting a column of values into rāSHio in one paste: File → Delimited List… The walkthrough pastes its own short demo list; the steps are the ones you run on your 35 prices.

Try It Now 7.4.2

Garrett Boone runs this lab but records the 35 cheapest listings in the paper rather than 35 chosen at random. His sample mean comes out near $285,000 and his standard deviation is much smaller than the one you will find in Table 7.4.1. Which of the two numbers is the more serious problem, and why does a confidence interval not fix either one?

Solution — a biased sample that looks precise

Step 1 — read what the low mean means. Taking the cheapest listings guarantees $\overline{x}$ sits below the true mean. That is bias: the error is in one direction and it does not shrink as he collects more data.

Step 2 — read what the small standard deviation means. The cheapest listings resemble each other, so $s$ is small. A small $s$ makes the error bound small and the interval narrow.

Step 3 — combine them. A narrow interval centered in the wrong place is worse than a wide one, because narrowness reads to a reader as precision. Garrett will report a tight interval that has essentially no chance of containing $\mu$.

Answer: the small standard deviation is the more dangerous of the two, because it disguises the bias as accuracy. A confidence interval fixes neither problem: every formula in this chapter assumes the sample was drawn at random, and when that assumption fails the interval is measuring the sampling error of a procedure that was aimed at the wrong target. There is no arithmetic repair for a bad sample — you go back and draw a new one.

7.4.3 Describe the Data

1. Compute the following.

a. $\overline{x} =$ ________

b. $s_{x} =$ ________

c. $n =$ ________

2. In words, define the random variable $\overline{X}$.

3. State the estimated distribution to use. Use both words and symbols.

Item 3 is the one to slow down on, because it is where the chapter's two halves separate. You know the population standard deviation $\sigma$ for exactly none of the quantities in this lab — nobody has ever computed the standard deviation of every home price in the county — so §7.1's normal-distribution interval does not apply. You have $s_x$, computed from your own 35 prices, and a sample standard deviation carries its own error that a normal distribution does not account for. That is what the Student t-distribution of §7.2 was built for, and the degrees of freedom are $n - 1 = 34$.

Try It Now 7.4.3

Marisol Peña has added up the demonstration data in Table 7.4.1 and hands you her two totals: the sum of the 35 prices is $14,350,000 and the sum of their squares is $\sum x^2 = 6,317,692 \times 10^{6}$. Compute $\overline{x}$, $s_x$, and $n$ from what she gives you.

Solution — the three summary numbers

Step 1 — count the values. Five rows of seven prices each gives $5 \times 7 = 35$, so $n = 35$.

Step 2 — divide the sum by the count.

$$ \overline{x} = \frac{14{,}350{,}000}{35} = 410{,}000 $$

Step 3 — find the total squared deviation.

$$ \sum (x - \overline{x})^2 = \sum x^2 - \frac{\left(\sum x\right)^2}{n} = (6{,}317{,}692 - 5{,}883{,}500) \times 10^{6} = 434{,}192 \times 10^{6} $$

Step 4 — divide by $n - 1$ and take the square root.

$$ s_x^2 = \frac{434{,}192 \times 10^{6}}{34} \approx 1.2770 \times 10^{10} \qquad s_x = \sqrt{1.2770 \times 10^{10}} \approx 113{,}006 $$

Answer: $\overline{x} = \mathbf{\$410{,}000}$, $s_x \approx \mathbf{\$113{,}006}$, and $n = \mathbf{35}$. Look at the size of that standard deviation before handing the numbers back to her — it is more than a quarter of the mean, which tells you the county's listings are spread across a very wide range of prices. That spread is what will make the interval as wide as it turns out to be, and it is a fact about the housing market, not a flaw in the sample.

Try it in rāSHio

With the 35 prices in a column, choose Stats → Summary Statistics in rāSHio to read the mean, the sample standard deviation and the count off one panel — exactly the three numbers item 1 of this subsection asks for. Work the sum-of-squares arithmetic by hand once, as the solution above does, then let the tool carry it for the rest of the lab.

Figure 7.4.3 — Reducing a column to its summary statistics in rāSHio: Stats → Summary Statistics. The walkthrough runs on its own demo scores; run it on your 35 prices and the same panel returns the three numbers the lab needs.

Try It Now 7.4.4

Define the random variable $\overline{X}$ in words for this study, and state the distribution you will use for it in both words and symbols.

Solution — naming the random variable and its model

Step 1 — say what varies. What changes from one repetition of the study to the next is not a single house price but the average of a fresh batch of 35.

Step 2 — write the definition. $\overline{X}$ is the mean sale price of a random sample of 35 homes recently listed in the county.

Step 3 — choose the model. The population standard deviation $\sigma$ is unknown, so $s_x$ stands in for it and the standardized sample mean follows a Student t-distribution with $n - 1 = 34$ degrees of freedom:

$$ \frac{\overline{X} - \mu}{s_x / \sqrt{n}} \sim t_{34} $$

Answer: in words, $\overline{X}$ is the mean sale price of a random sample of 35 recently listed homes; in symbols the study uses the Student t-distribution with 34 degrees of freedom, written $t_{34}$. The most common error here is defining $\overline{X}$ as "the sale price of a home" — that is $X$, the individual price, and it is not what this interval is about. Every number in the rest of the lab is a statement about the average of 35 prices, and the two have very different amounts of spread.

7.4.4 Find the Confidence Interval

Definition 7.4.1: Error Bound for a Population Mean (t-based)

For a sample of size $n$ with sample mean $\overline{x}$ and sample standard deviation $s_x$, drawn from a population whose standard deviation is unknown, the error bound for the mean at confidence level $CL$ is

$$ EBM = t_{\alpha/2} \cdot \frac{s_x}{\sqrt{n}} $$

where $\alpha = 1 - CL$ and $t_{\alpha/2}$ is the value that puts an area of $\alpha/2$ in the right tail of the Student t-distribution with $n - 1$ degrees of freedom. The confidence interval is then $(\overline{x} - EBM,\ \overline{x} + EBM)$.

The error bound is a distance, not a place

$EBM$ answers one question — how far from your sample mean do you have to reach before you have covered enough of the t-distribution? The interval is just that distance laid off in both directions from $\overline{x}$.

This is Definition 7.2.3 from §7.2, restated here because calculating the interval and the error bound is the step of the lab students most often try to do from memory.

One piece of that formula is worth naming on its own, because it is the piece you will reuse in every row of the table at the end of the lab.

Definition 7.4.2: Standard Error of the Mean

For a sample of size $n$ with sample standard deviation $s_x$, the standard error of the mean is

$$ SE = \frac{s_x}{\sqrt{n}} $$

It measures how much the sample mean $\overline{X}$ varies from sample to sample, and it depends only on the data — not on the confidence level.

Computing it once first will save you arithmetic all the way through §7.4.6. Everything the confidence level touches is packed into the single multiplier $t_{\alpha/2}$, so the last part of this lab can fill an entire table of intervals without ever recomputing $\overline{x}$ or $s_x$: you find the standard error once, then multiply it by a different t-value for each row.

1. Calculate the confidence interval and the error bound.

a. Confidence Interval: ________

b. Error Bound: ________

2. How much area is in both tails combined? $\alpha =$ ________

3. How much area is in each tail? $\frac{\alpha}{2} =$ ________

4. Fill in the blanks on the graph with the area in each section. Then fill in the number line with the upper and lower limits of the confidence interval and the sample mean.

Sketch the graph yourself rather than picturing it. Draw a t-curve, shade the middle and label it with the confidence level, shade the two tails and label each with $\alpha/2$, then run a number line underneath with $\overline{x}$ at the center and the two interval endpoints below the boundaries of the shaded middle. The picture is doing real work: it is the only place where the confidence level, the two tail areas, and the two endpoints all appear at once, and drawing it is how you catch a sign error or a misread t-value before it reaches your write-up.

Try It Now 7.4.5

Using $\overline{x} = \$410,000$, $s_x = \$113,006$, and $n = 35$ from Try It Now 7.4.3, find the error bound and the 90% confidence interval for the mean sale price. Use $t_{0.05} = 1.6909$ with 34 degrees of freedom.

Solution — the 90% interval for mean sale price

Step 1 — compute the standard error.

$$ \frac{s_x}{\sqrt{n}} = \frac{113{,}006}{\sqrt{35}} = \frac{113{,}006}{5.9161} \approx 19{,}101 $$

Step 2 — multiply by the t-value.

$$ EBM = 1.6909 \times 19{,}101 \approx 32{,}298 $$

Step 3 — lay that distance off in both directions from the sample mean.

$$ 410{,}000 - 32{,}298 = 377{,}702 \qquad 410{,}000 + 32{,}298 = 442{,}298 $$

Answer: the error bound is about $\mathbf{\$32{,}298}$ and the 90% confidence interval runs from $\mathbf{\$377{,}702}$ to $\mathbf{\$442{,}298}$. Notice how wide that is — roughly $65,000 from end to end — and notice where the width came from. It is not sloppiness in the sample. It is the $113,006 standard deviation divided by the square root of only 35 homes. To halve the width the class would need four times as many listings, because the $\sqrt{n}$ in the denominator means precision improves with the square root of effort, not with effort itself.

Try it in rāSHio

The whole interval comes out in one step. In rāSHio choose Stats → T Stats, tick the Confidence Interval option rather than running a hypothesis test, and enter this lab’s summary values — 410000 for the sample mean, 113006 for the sample standard deviation, 35 for the sample size and 0.90 for the confidence level. It returns the same interval the three steps above build by hand, which makes it a way to check your work rather than a way to skip it.

Figure 7.4.4 — Building the t-interval in one panel: Stats → T Stats in rāSHio. The clip runs the panel’s hypothesis-test mode on its own demo numbers; the confidence-interval option takes the same four fields.

Try It Now 7.4.6

For that same 90% interval, state $\alpha$ and $\frac{\alpha}{2}$, and say which three numbers you would write on the number line beneath the shaded t-curve.

Solution — tail areas and the labelled sketch

Step 1 — take alpha from the confidence level.

$$ \alpha = 1 - CL = 1 - 0.90 = 0.10 $$

Step 2 — split it between the two tails. The t-distribution is symmetric, so the leftover area divides evenly.

$$ \frac{\alpha}{2} = \frac{0.10}{2} = 0.05 $$

Step 3 — label the sketch. The shaded middle carries $CL = 0.90$; each tail carries 0.05; the number line beneath carries the lower limit, the sample mean, and the upper limit.

Answer: $\alpha = \mathbf{0.10}$ combined and $\frac{\alpha}{2} = \mathbf{0.05}$ in each tail, and the three numbers on the number line are $\mathbf{\$377{,}702}$, $\mathbf{\$410{,}000}$, and $\mathbf{\$442{,}298}$. The three areas have to total 1: $0.05 + 0.90 + 0.05 = 1.00$. That check catches the single most common mistake in this step, which is putting the full $\alpha$ in each tail instead of half of it — a slip that would send you to the t-value for an 80% interval while your paper still says 90%.

5. Some students think that a 90% confidence interval contains 90% of the data. Use your list of prices and count how many of the values lie within your confidence interval. What percent is this? Is this percent close to 90%? Explain why this percent should or should not be close to 90%.

Try It Now 7.4.7

Hannah Brightwell is convinced that a 90% confidence interval should hold about 90% of the prices, and she wants to see it checked. Count how many of the 35 prices in Table 7.4.1 fall inside the interval $(\$377,702,\ \$442,298)$, convert that to a percent, and explain to her why the answer is nowhere near 90%.

Solution — counting data inside the interval

Step 1 — find the values in range. Reading Table 7.4.1 from $377,702 upward and stopping at $442,298:

$$ 379{,}000,\ 385{,}000,\ 392{,}000,\ 399{,}000,\ 405{,}000,\ 412{,}000,\ 420{,}000,\ 429{,}000,\ 438{,}000 $$

Step 2 — count them and convert to a percent. That is 9 values out of 35.

$$ \frac{9}{35} \approx 0.257 = 25.7\% $$

Step 3 — say why the two percentages are unrelated. The interval was built around $\overline{X}$, whose spread is the standard error $\frac{s_x}{\sqrt{n}} \approx \$19,101$. Individual prices spread out by $s_x \approx \$113,006$, nearly six times as much.

Answer: 9 of the 35 prices, or about 25.7%, lie inside the interval, and it should not be close to 90% — so her expectation was the wrong one. The interval is a statement about where the population mean plausibly sits, not about where individual homes sell. Because sample means vary about six times less than individual prices do, an interval built to capture the mean is far too narrow to capture most of the data — and making it wide enough to hold 90% of the homes would tell you almost nothing about $\mu$. A range that did describe individual prices is a different tool with a different name, a prediction interval, and it is much wider than this one.

Try it in rāSHio

Counting how many prices fall inside the interval is a hunt on an unsorted column and a glance on a sorted one. Select your 35 prices in rāSHio, choose Transform → Sort to put them in ascending order, then read off the unbroken block of values sitting between the two endpoints — the length of that block is the count item 5 asks for.

Figure 7.4.5 — Putting a column in ascending order so the in-interval count is a glance rather than a hunt: Transform → Sort in rāSHio. The walkthrough sorts its own demo values; the steps are the ones you run on your prices.

7.4.5 Describe the Confidence Interval

1. In two to three complete sentences, explain what a confidence interval means in general, as if you were talking to someone who has not taken statistics.

2. In one to two complete sentences, explain what this confidence interval means for this particular study.

The confidence belongs to the method

Think of the procedure as a ring toss where the peg is fixed and you are the one moving. Ninety percent of your throws land the ring over the peg — but once a ring is on the ground, it either encircles the peg or it does not.

Item 1 is asking for the general sentence and item 2 is asking for the specific one, and they fail in different ways. The general sentence goes wrong when it attaches the probability to the interval instead of to the procedure: "there is a 90% chance the true mean is between these two numbers" sounds right and is the error the whole chapter is built to prevent, because $\mu$ is a fixed number and your two endpoints are fixed numbers, so no chance remains once they are written down. Say instead that the method captures the true mean 90% of the time. The specific sentence goes wrong in the other direction, by being too vague to check — "we are 90% confident about home prices" names no population, no quantity, and no interval. A good specific sentence names all three: who was studied, what was averaged, and which two numbers came out. Write both, then read them to someone who has not taken the course and see whether they can tell you what you measured.

Try It Now 7.4.8

Rafa Solano and their partner are house-hunting in the county and have asked what the class's interval actually tells them. Write the two sentences the lab asks for, using the demonstration data's 90% interval of $(\$377,702,\ \$442,298)$. Then say what is wrong with telling them "there is a 90% chance the mean home price is between $377,702 and $442,298."

Solution — writing both interpretations

Step 1 — the general sentence. A confidence interval is a range built from a sample by a recipe that has a known success rate. If you repeated the whole study many times, each sample producing its own interval, about 90% of those intervals would contain the true population value. Any single interval either contains it or misses it, and you never find out which.

Step 2 — the specific sentence. We are 90% confident that the true mean sale price of all homes recently listed in Butte County lies between $377,702 and $442,298.

Step 3 — diagnose the bad sentence. It treats $\mu$ as a random quantity that might fall in a fixed range. $\mu$ is a constant, and once the endpoints are computed they are constants too, so the statement is either entirely true or entirely false — there is no 90% about it.

Answer: the two sentences above are what Rafa and their partner should be told. The bad sentence is wrong because the randomness was in the sampling, not in the population mean: before you drew the sample, the procedure had a 90% chance of producing an interval that covers $\mu$; afterward, the dice have already been rolled. Saying "we are 90% confident" instead of "there is a 90% chance" is not word-fussing — it is the difference between describing a method's track record and making a probability claim about a fixed number.

7.4.6 Use the Data to Construct Confidence Intervals

1. Using the given information, construct a confidence interval for each confidence level given.

**Table 7.4.3** — Error bound and confidence interval for the mean sale price at four confidence levels, computed from your own sample.
Confidence level	EBM / Error Bound	Confidence Interval
50%
80%
95%
99%

2. What happens to the EBM as the confidence level increases? Does the width of the confidence interval increase or decrease? Explain why this happens.

The four rows share everything except one number. Your $\overline{x}$, your $s_x$, and your $n$ are fixed by the sample you already collected, so the standard error $\frac{s_x}{\sqrt{n}}$ is the same in every row — compute it once, write it in the margin, and reuse it. All that changes down the column is $t_{\alpha/2}$, and it changes because raising the confidence level means demanding more area under the middle of the t-curve, which forces the boundaries further out into the tails. Filling this table is really just multiplying one fixed number by four different multipliers, and seeing that plainly is the point of the exercise.

Try It Now 7.4.9

Tyler Ashgrove and his husband want to know how much the interval moves if the class demands more confidence than 90%. Complete Table 7.4.3 for the demonstration data, using the standard error $\frac{s_x}{\sqrt{n}} \approx \$19,101$ and the t-values $t_{0.25} = 0.6818$, $t_{0.10} = 1.3070$, $t_{0.025} = 2.0322$, and $t_{0.005} = 2.7284$, all with 34 degrees of freedom. Then answer item 2 for him.

Solution — four confidence levels compared

Step 1 — multiply the standard error by each t-value.

$$ 0.6818 \times 19{,}101 \approx 13{,}023 \qquad 1.3070 \times 19{,}101 \approx 24{,}965 $$ $$ 2.0322 \times 19{,}101 \approx 38{,}817 \qquad 2.7284 \times 19{,}101 \approx 52{,}115 $$

Step 2 — lay each error bound off from $\overline{x} = \$410,000$.

**Table 7.4.4** — The demonstration data's error bound and confidence interval at each of the four confidence levels.
Confidence level	EBM	Confidence Interval	Width
50%	$13,023	($396,977, $423,023)	$26,046
80%	$24,965	($385,035, $434,965)	$49,930
95%	$38,817	($371,183, $448,817)	$77,634
99%	$52,115	($357,885, $462,115)	$104,230

Step 3 — read the trend. The error bound grows from about $13,000 to about $52,115, and the width grows right along with it — the 99% interval is four times as wide as the 50% one.

Answer: as the confidence level increases, the EBM increases and the interval gets wider. Higher confidence means insisting that more of the t-curve's area sits inside the interval, which pushes the cutoffs further into the tails and makes $t_{\alpha/2}$ larger; the standard error never moved. The trade is unavoidable and it is worth stating out loud in your write-up: certainty and precision pull against each other. The 99% interval is very likely to contain $\mu$ and spans over $100,000, which is almost useless to a buyer like him; the 50% interval pins the mean to a $26,000 range and is wrong about half the time. Ninety and ninety-five percent are conventions because they sit in the usable middle, not because there is anything mathematically special about them.

Try it in rāSHio

Every row of the table above needs a different multiplier at the same 34 degrees of freedom. In rāSHio choose Distributions → T, set the degrees of freedom to 34, and read the critical value beside the Confidence field for each of 0.50, 0.80, 0.95 and 0.99 — the standard error you computed once never moves, so those four numbers are the only new arithmetic the table needs.

Figure 7.4.6 — Reading a critical value off the t-panel in rāSHio: Distributions → T. The walkthrough settles on its own demo case, df 12 at 0.95; set the degrees of freedom to 34 and step the Confidence field through 0.50, 0.80, 0.95 and 0.99 to get the four multipliers the table needs.

Key Terms

error bound for a population mean — written $EBM$, the distance from the sample mean to either endpoint of the confidence interval; $EBM = t_{\alpha/2} \cdot \frac{s_x}{\sqrt{n}}$.

standard error of the mean — the standard deviation of the sample mean, $\frac{s_x}{\sqrt{n}}$; the part of the error bound that does not change when the confidence level does.