8.1 Null and Alternative Hypotheses
SLO 4
Demonstrate an understanding of, and ability to use, basic ideas of statistical processes, including hypothesis tests and confidence interval estimation.
Every hypothesis test in this chapter starts here: you write the null and the alternative before you touch the data. Getting that pair right — equality in the null, the direction word in the alternative — is what makes the arithmetic later answer the question you meant to ask.
SLO 5
Identify appropriate statistical techniques and use technology-based statistical analysis to describe, interpret, and communicate results.
No calculator work happens in this section, but it sets up the communicating half of the outcome: it fixes the wording of the two decisions a test can reach, and why 'do not reject' is never the same sentence as 'accept.'
Learning Objectives
By the end of this section, you will be able to:
- state the null hypothesis and the alternative hypothesis for a claim about a population mean or a population proportion;
- choose the correct symbol for each hypothesis from the wording of the claim;
- explain why the null hypothesis always carries an equality and the alternative never does;
- describe the two decisions a hypothesis test can end in, and say what each one does and does not prove.
Everything in this chapter builds on one small piece of writing: a pair of sentences you set down before you look at the data. A hypothesis test does not begin with a formula or a calculator. It begins by considering two hypotheses — one called the null hypothesis and the other the alternative hypothesis — which contain opposing viewpoints and cannot both be true. Writing both down forces you to say, in advance, what would count as an interesting result and what would count as business as usual. Get the pair wrong and every calculation after it answers the wrong question, no matter how clean the arithmetic is.
8.1.1 The Two Hypotheses
The null hypothesis, written \(H_{0}\), is a statement of no difference between the variables — they are not related. This can often be considered the status quo, and as a result, if you cannot accept the null it requires some action.
Definition 8.1.1 - The null hypothesis always contains the claimed value, whichever of the three forms it takes.
The alternative hypothesis, written \(H_{a}\), is a claim about the population that is contradictory to \(H_{0}\), and what we conclude when we reject \(H_{0}\). This is usually what the researcher is trying to prove.
A hypothesis test runs like a courtroom. The null hypothesis is the defendant, presumed true until the evidence says otherwise. The sample data is the evidence, and rejecting the null is the verdict that the evidence was strong enough.
Definition 8.1.2 - The alternative hypothesis takes exactly what the null leaves over, and stops short of the boundary.
Since the null and alternative hypotheses are contradictory, you must examine evidence to decide whether you have enough of it to reject the null hypothesis or not. The evidence is in the form of sample data.
After you have determined which hypothesis the sample supports, you make a decision, and there are exactly two options. You reject \(H_{0}\) if the sample information favors the alternative hypothesis. You do not reject \(H_{0}\) — some books say "decline to reject \(H_{0}\)" — if the sample information is insufficient to reject the null hypothesis. Notice what is missing from that pair: you never accept \(H_{0}\). Failing to find enough evidence against a claim is not the same as showing the claim is true. A jury that acquits has not declared the defendant innocent, only that the prosecution did not meet its burden. The same asymmetry shows up all through statistics, and it comes from where the burden of proof sits: the whole procedure is built to protect the null, so the only thing a test can ever do is find the evidence against it strong or find it weak. Weak evidence tells you the sample did not settle the question, which is a very different sentence from "the claim is true." Keeping that wording straight from the very first section is worth more than it looks, because the phrase you write in your conclusion is the part a reader checks first, and it is the one place where a correct calculation can still earn a wrong answer.
A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.
Solution — hypotheses for a cholesterol trial
Step 1 — Name the parameter. The claim is about the percentage of cholesterol reduction in the whole population of patients, so the parameter is a proportion, \(p\), and the claimed value is 0.25.
Step 2 — Write the status quo as the null. The trial exists to check the 25% figure, so the null says the reduction is exactly what was claimed:
$$ H_{0}: p = 0.25 $$Step 3 — Write the contradiction as the alternative. "Whether or not" gives no direction, so the alternative is simply that the reduction is not 25%:
$$ H_{a}: p \neq 0.25 $$Answer: \(H_{0}: p = 0.25\) and \(H_{a}: p \neq 0.25\).
Marcy Ellingson and her wife both volunteer at their Santa Clara County polling place. Marcy wants to test the claim that more than 30% of the registered voters in the county voted in the primary election. State the null and alternative hypotheses for her test.
Solution — voter turnout above 30%
Step 1 — Identify the parameter. Turnout is a share of registered voters, so the parameter is a population proportion \(p\).
Step 2 — Put the status quo in \(H_{0}\). The claim being tested is that turnout is high, so the null is the opposite position: no more than 30% of the registered voters voted.
$$ H_{0}: p \leq 0.30 $$Step 3 — Put the researcher's claim in \(H_{a}\). More than 30% of the registered voters voted in the primary election.
$$ H_{a}: p > 0.30 $$Answer: \(H_{0}: p \leq 0.30\) and \(H_{a}: p > 0.30\).
8.1.2 Choosing the Symbols
Once you know which claim is the null and which is the alternative, the only thing left is picking the symbol for each. There are three legal pairings, and they come straight from how the claim is worded.
| \(H_{0}\) | \(H_{a}\) |
|---|---|
| equal \((=)\) | not equal \((\neq)\) or greater than \((>)\) or less than \((<)\) |
| greater than or equal to \((\geq)\) | less than \((<)\) |
| less than or equal to \((\leq)\) | more than \((>)\) |
Read the table across, not down. Each row is a matched pair, and the two entries in a row always cover every possible value of the parameter between them with no overlap. That is the property you are really after: the null and the alternative have to partition the number line, so that whatever the truth is, exactly one of your two sentences contains it.
\(H_{0}\) always has a symbol with an equal in it. \(H_{a}\) never has a symbol with an equal in it. The choice among \(=\), \(\leq\), and \(\geq\) depends on the wording of the hypothesis test.
Be aware that many researchers use \(=\) in the null hypothesis even when the alternative carries \(>\) or \(<\). This practice is acceptable, because we only ever make the decision to reject or not reject the null hypothesis, and that decision turns on the boundary value — the single number where the two claims meet. Whether you write \(H_{0}: \mu \geq 5\) or \(H_{0}: \mu = 5\), the arithmetic you are about to do uses \(\mu = 5\) either way, because that is the value least favorable to the null that the null still allows. So both forms lead to the same test statistic and the same conclusion. Your instructor or your textbook will settle on one convention; the useful habit is to notice which one is in front of you and stay consistent inside a single problem, rather than to argue that one is right. What is never acceptable is an equal sign in the alternative, because then the two claims would overlap and no sample could tell them apart.
We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol \((=, \neq, \geq, <, \leq, >)\) for the null and alternative hypotheses.
a) \(H_{0}: \mu \, \underline{\hspace{2em}} \, 66\)
b) \(H_{a}: \mu \, \underline{\hspace{2em}} \, 66\)
Solution — symbols for an eighth-grade height claim
Step 1 — Read the direction out of the wording. "Whether the mean height is 66 inches" asks only whether the claim holds, not whether heights run high or low. There is no direction, so this is a two-sided test.
Step 2 — Fill in the symbols. The null takes the equality and the alternative takes its negation:
$$ H_{0}: \mu = 66 \qquad H_{a}: \mu \neq 66 $$Answer: a. \(=\) b. \(\neq\)
Rene Quintero (they/them) runs the queer student union's academic-support program, and they want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). State the null and alternative hypotheses for their test.
Solution — testing a mean GPA of 2.0
Step 1 — Identify the parameter. A GPA averaged over a population of students is a population mean, \(\mu\).
Step 2 — Write the null. "Different from 2.0" gives no direction, so the null is the plain equality:
$$ H_{0}: \mu = 2.0 $$Step 3 — Write the alternative. The contradiction of "equals 2.0" is "does not equal 2.0":
$$ H_{a}: \mu \neq 2.0 $$Answer: \(H_{0}: \mu = 2.0\) and \(H_{a}: \mu \neq 2.0\).
8.1.3 Writing Hypotheses from a Claim
Most of the work in this section is translation. You are handed an English sentence about a population and you have to turn it into two mathematical statements. Three questions get you there every time: What is the parameter — a mean \(\mu\) or a proportion \(p\)? What number is being claimed? And does the claim point in a direction, or does it just say "different"?
Whatever the researcher hopes to show goes in \(H_{a}\). If the sentence says "we want to test if fewer than…", "more than…", or "has increased", that phrase is the alternative, word for word.
Direction words are the part students misread most often, so it pays to slow down on them. "Fewer than", "less than", "under", and "has decreased" all point left, and they belong in \(H_{a}\) with a \(<\). "More than", "greater than", "over", "exceeds", and "has increased" all point right, and they belong in \(H_{a}\) with a \(>\). "Different from", "not equal to", "has changed", and "whether or not" point nowhere in particular, and they get \(\neq\). Once the alternative is written, the null is forced — it is whatever is left over, which means it always includes the boundary value and always carries an equality. Working in that order, alternative first and null second, removes almost all of the guesswork, because the English sentence usually hands you the alternative directly and almost never hands you the null.
We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol \((=, \neq, \geq, <, \leq, >)\) for the null and alternative hypotheses.
a) \(H_{0}: \mu \, \underline{\hspace{2em}} \, 45\)
b) \(H_{a}: \mu \, \underline{\hspace{2em}} \, 45\)
Solution — symbols for a lesson-plan time claim
Step 1 — Find the alternative. "Fewer than 45 minutes" points left, so \(H_{a}: \mu < 45\).
Step 2 — Take the leftover as the null. Everything the alternative does not cover is 45 minutes or more:
$$ H_{0}: \mu \geq 45 \qquad H_{a}: \mu < 45 $$Answer: a. \(\geq\) b. \(<\)
We want to test if college students take less than five years to graduate from college, on the average. State the null and alternative hypotheses.
Solution — time to graduate under five years
Step 1 — Identify the parameter. "On the average" over all college students is a population mean \(\mu\), measured in years.
Step 2 — Write the alternative from the direction word. "Less than five years" points left:
$$ H_{a}: \mu < 5 $$Step 3 — Write the null as the complement. The null holds the boundary and everything above it:
$$ H_{0}: \mu \geq 5 $$Answer: \(H_{0}: \mu \geq 5\) and \(H_{a}: \mu < 5\).
On a state driver's test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol \((=, \neq, \geq, <, \leq, >)\) for the null and alternative hypotheses.
a) \(H_{0}: p \, \underline{\hspace{2em}} \, 0.40\)
b) \(H_{a}: p \, \underline{\hspace{2em}} \, 0.40\)
Solution — symbols for a driver's-test pass rate
Step 1 — Find the alternative. "More than 40% pass" points right, and a pass rate is a proportion, so \(H_{a}: p > 0.40\).
Step 2 — Take the leftover as the null. Everything at or below the boundary:
$$ H_{0}: p \leq 0.40 \qquad H_{a}: p > 0.40 $$Answer: a. \(\leq\) b. \(>\)
In an issue of U.S. News and World Report, an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test whether the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.
Solution — U.S. share taking AP exams
Step 1 — Strip out the numbers you do not need. The figures for France, Germany, and Israel, and the 4.4% pass rate, are background. The claim under test is about the 6.6% who take the exams.
Step 2 — Identify the parameter. A percentage of all U.S. students is a population proportion \(p\), and \(6.6\% = 0.066\).
Step 3 — Write the alternative, then the null. "More than 6.6%" points right:
$$ H_{0}: p \leq 0.066 \qquad H_{a}: p > 0.066 $$Answer: \(H_{0}: p \leq 0.066\) and \(H_{a}: p > 0.066\).
Try it in rāSHio
Open rāSHio and choose Stats → Prop Stats: the dialog is built around the pair you just wrote. The claimed value goes in its Ho: p = field, and the Ha: row beside it holds the direction. That row starts at \(
eq\), so the clip below runs a two-tailed test — for a one-sided claim like Example 8.1.4's "more than", you change it to \(>\) yourself before calculating.
Figure 8.1.1 — A proportion test in rāSHio: the claimed value is typed into the Ho field while the Ha row keeps its default \(
eq\), so this run is two-tailed.
Bring to class a newspaper, some news magazines, and some Internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.
Problem Set 8.1
Problem 1. You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. What is the random variable? Describe in words.
Problem 1 Solution
Step 1 — Ask what is being measured: The claim is about the speed of your cable Internet connection, averaged over many measurements, not about a single reading.
Step 2 — Name the random variable in words: Because the claim is about an average, the random variable is the sample mean of that measurement.
Answer: The random variable is the mean Internet speed in Megabits per second.
Problem 2. You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. State the null and alternative hypotheses.
Problem 2 Solution
Step 1 — Find the direction word: "More than three Megabits per second" points right, so that claim is the alternative.
Step 2 — Write the alternative, then take the complement as the null: The parameter is a population mean \(\mu\), measured in Megabits per second.
$$ H_{0}: \mu \leq 3 \qquad H_{a}: \mu > 3 $$Answer: \(H_{0}: \mu \leq 3\) and \(H_{a}: \mu > 3\).
Problem 3. The American family has an average of two children. What is the random variable? Describe in words.
Problem 3 Solution
Step 1 — Ask what is being measured: "The American family has an average of two children" is a claim about a mean count of children across all American families.
Step 2 — Name the random variable in words: Since the claim is about an average, the random variable is the sample mean of that count.
Answer: The random variable is the mean number of children an American family has.
Problem 4. The mean entry level salary of an employee at a company is $58,000. You believe it is higher for IT professionals in the company. State the null and alternative hypotheses.
Problem 4 Solution
Step 1 — Identify the parameter: An entry level salary averaged over employees is a population mean \(\mu\), measured in dollars, with claimed value $58,000.
Step 2 — Put the belief in the alternative: You believe the mean is higher for IT professionals, and "higher" points right.
$$ H_{0}: \mu \leq \$58,000 \qquad H_{a}: \mu > \$58,000 $$Step 3 — Note the equivalent form: Writing \(H_{0}: \mu = \$58,000\) is also accepted, since the test statistic is computed at the boundary value $58,000 either way.
Answer: \(H_{0}: \mu \leq \$58,000\) and \(H_{a}: \mu > \$58,000\).
Problem 5. Dr. Minho Kang, a sociologist, claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test his claim to see if the proportion is actually less. What is the random variable? Describe in words.
Problem 5 Solution
Step 1 — Ask what is being measured: A "probability that a person picked at random is visiting" is a share of the people in Times Square, not an average of a numeric measurement.
Step 2 — Name the random variable in words: A share is a proportion, so the random variable is the sample proportion.
Answer: The random variable is the proportion of people picked at random in Times Square visiting the city.
Problem 6. Dr. Minho Kang, a sociologist, claims the probability that a person picked at random in Times Square in New York City is visiting the area is 0.83. You want to test to see if his claim is correct. State the null and alternative hypotheses.
Problem 6 Solution
Step 1 — Identify the parameter: The claim is about a population proportion \(p\), with claimed value 0.83.
Step 2 — Read the direction: "Test to see if the claim is correct" asks only whether 0.83 holds. There is no direction word, so this is a two-sided test.
$$ H_{0}: p = 0.83 \qquad H_{a}: p \neq 0.83 $$Answer: \(H_{0}: p = 0.83\) and \(H_{a}: p \neq 0.83\).
Problem 7. In a population of fish, approximately 42% are female. A test is conducted to see if, in fact, the proportion is less. State the null and alternative hypotheses.
Problem 7 Solution
Step 1 — Identify the parameter: The share of a fish population that is female is a population proportion \(p\), with claimed value \(42\% = 0.42\).
Step 2 — Read the direction: The test is run "to see if, in fact, the proportion is less," which points left, so that claim is the alternative.
$$ H_{0}: p = 0.42 \qquad H_{a}: p < 0.42 $$Answer: \(H_{0}: p = 0.42\) and \(H_{a}: p < 0.42\).
Problem 8. Suppose that a recent article stated that the mean time spent in jail by a first-time convicted burglar is 2.5 years. A study was then done to see if the mean time has increased in the new century. A random sample of 26 first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was 3 years with a standard deviation of 1.8 years. Suppose that it is somehow known that the population standard deviation is 1.5. If you were conducting a hypothesis test to determine if the mean length of jail time has increased, what would the null and alternative hypotheses be? The distribution of the population is normal.
a) \(H_{0}\): \(\underline{\hspace{6em}}\)
b) \(H_{a}\): \(\underline{\hspace{6em}}\)
Problem 8 Solution
Step 1 — Sort the numbers you need from the ones you do not: The sample size 26, the sample mean of 3 years, the sample standard deviation of 1.8, and the population standard deviation of 1.5 all belong to the calculation that comes later. The hypotheses use only the claimed value, 2.5 years.
Step 2 — Identify the parameter: Time in jail averaged over all first-time convicted burglars is a population mean \(\mu\), in years.
Step 3 — Read the direction: The study was done "to see if the mean time has increased," which points right, so that is the alternative.
$$ H_{0}: \mu \leq 2.5 \qquad H_{a}: \mu > 2.5 $$Answer: a. \(H_{0}: \mu \leq 2.5\) b. \(H_{a}: \mu > 2.5\)
Problem 9. A random survey of 75 student loan recipients revealed that the mean length of time in repayment is 17.4 years with a standard deviation of 6.3 years. If you were conducting a hypothesis test to determine if the population mean time in repayment could likely be 15 years, what would the null and alternative hypotheses be?
a) \(H_{0}\): \(\underline{\hspace{6em}}\)
b) \(H_{a}\): \(\underline{\hspace{6em}}\)
Problem 9 Solution
Step 1 — Identify the parameter: Length of time in repayment, averaged over all student loan recipients, is a population mean \(\mu\), in years, with claimed value 15.
Step 2 — Read the direction: "Could likely be 15 years" asks only whether the claim holds, with no direction, so this is a two-sided test. The sample mean of 17.4 and standard deviation of 6.3 belong to the later calculation, not to the hypotheses.
$$ H_{0}: \mu = 15 \qquad H_{a}: \mu \neq 15 $$Answer: a. \(H_{0}: \mu = 15\) b. \(H_{a}: \mu \neq 15\)
Problem 10. The National Institute of Mental Health published an article stating that in any one-year period, approximately 9.5% of American adults suffer from depression or a depressive illness. Suppose that in a survey of 100 people in a certain town, seven of them suffered from depression or a depressive illness. If you were conducting a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population, what would the null and alternative hypotheses be?
a) \(H_{0}\): \(\underline{\hspace{6em}}\)
b) \(H_{a}\): \(\underline{\hspace{6em}}\)
Problem 10 Solution
Step 1 — Identify the parameter: The share of adults suffering from depression is a population proportion \(p\), with claimed value \(9.5\% = 0.095\).
Step 2 — Read the direction: The test asks whether the true proportion in that town is lower than the national figure, which points left.
$$ H_{0}: p = 0.095 \qquad H_{a}: p < 0.095 $$Step 3 — Set aside the sample: The 7 out of 100 townspeople give a sample proportion of 0.07; that number is evidence for the test, not part of the hypotheses.
Answer: a. \(H_{0}: p = 0.095\) b. \(H_{a}: p < 0.095\)
Problem 11. Some of the following statements refer to the null hypothesis, some to the alternate hypothesis. State the null hypothesis, \(H_{0}\), and the alternative hypothesis, \(H_{a}\), in terms of the appropriate parameter (\(\mu\) or \(p\)).
a) The mean number of years Americans work before retiring is 34.
b) At most 60% of Americans vote in presidential elections.
c) The mean starting salary for San Jose State University graduates is at least $100,000 per year.
d) Twenty-nine percent of high school seniors get drunk each month.
e) Fewer than 5% of adults ride the bus to work in Los Angeles.
f) The mean number of cars a person owns in her lifetime is not more than ten.
g) About half of Americans prefer to live away from cities, given the choice.
h) Europeans have a mean paid vacation each year of six weeks.
i) The chance of developing breast cancer is under 11% for females.
j) Private universities' mean tuition cost is more than $20,000 per year.
Problem 11 Solution
Step 1 — Decide the parameter for each statement: A statement about an average uses a population mean \(\mu\); a statement about a percentage or a chance uses a population proportion \(p\).
Step 2 — Put the equality in the null and the direction word in the alternative: Taking each statement in turn:
a. \(H_{0}: \mu = 34\); \(H_{a}: \mu \neq 34\)
b. \(H_{0}: p \leq 0.60\); \(H_{a}: p > 0.60\)
c. \(H_{0}: \mu \geq 100,000\); \(H_{a}: \mu < 100,000\)
d. \(H_{0}: p = 0.29\); \(H_{a}: p \neq 0.29\)
e. \(H_{0}: p = 0.05\); \(H_{a}: p < 0.05\)
f. \(H_{0}: \mu \leq 10\); \(H_{a}: \mu > 10\)
g. \(H_{0}: p = 0.50\); \(H_{a}: p \neq 0.50\)
h. \(H_{0}: \mu = 6\); \(H_{a}: \mu \neq 6\)
i. \(H_{0}: p = 0.11\); \(H_{a}: p < 0.11\)
j. \(H_{0}: \mu = 20,000\); \(H_{a}: \mu > 20,000\)
Step 3 — Check the pattern: Statements with no direction ("is 34", "is six weeks") give a two-sided pair. Statements carrying a direction word ("at most", "at least", "fewer than", "not more than", "under", "more than") put that direction in \(H_{a}\), and the null takes whatever is left, including the boundary value.
Answer: As listed in Step 2. Values in parts c and j are dollars per year.
Problem 12. Over the past few decades, public health officials have examined the link between weight concerns and teen girls' smoking. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). After four years the girls were surveyed again. Sixty-three said they smoked to stay thin. Is there good evidence that more than 30% of the teen girls smoke to stay thin? The alternative hypothesis is:
a) \(p < 0.30\)
b) \(p \leq 0.30\)
c) \(p \geq 0.30\)
d) \(p > 0.30\)
Problem 12 Solution
Step 1 — Find what is being claimed: The researchers want evidence that more than 30% of the teen girls smoke to stay thin.
Step 2 — Put that claim in the alternative: "More than 30%" points right, so \(H_{a}: p > 0.30\). Options b and c carry an equality, which an alternative never does, and option a points the wrong way.
Answer: d. \(p > 0.30\)
Problem 13. A statistics instructor believes that fewer than 20% of Evergreen Valley College (EVC) students attended the opening night midnight showing of the latest Marvel Universe movie. The instructor surveys 84 students and finds that 11 attended the midnight showing. An appropriate alternative hypothesis is:
a) \(p = 0.20\)
b) \(p > 0.20\)
c) \(p < 0.20\)
d) \(p \leq 0.20\)
Problem 13 Solution
Step 1 — Find what is being claimed: The instructor believes fewer than 20% of EVC students attended the midnight showing.
Step 2 — Put that claim in the alternative: "Fewer than 20%" points left, so \(H_{a}: p < 0.20\). Options a and d carry an equality, which belongs in the null, and option b points the wrong way. The survey result of 11 out of 84 is evidence for the test, not part of the hypotheses.
Answer: c. \(p < 0.20\)
Problem 14. Previously, an organization reported that teenagers spent 4.5 hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. Fifteen randomly chosen teenagers were asked how many hours per week they spend on the phone. The sample mean was 4.75 hours with a sample standard deviation of 2.0. Conduct a hypothesis test. The null and alternative hypotheses are:
a) \(H_{0}: \overline{x} = 4.5\), \(H_{a}: \overline{x} > 4.5\)
b) \(H_{0}: \mu \geq 4.5\), \(H_{a}: \mu < 4.5\)
c) \(H_{0}: \mu = 4.75\), \(H_{a}: \mu > 4.75\)
d) \(H_{0}: \mu = 4.5\), \(H_{a}: \mu > 4.5\)
Problem 14 Solution
Step 1 — Check the symbol in each option: Hypotheses are always statements about a population parameter. Option a uses \(\overline{x}\), the sample mean, so it is out immediately.
Step 2 — Check the claimed value: The organization's reported figure is 4.5 hours. Option c builds its hypotheses around 4.75, which is the sample mean from the new data, not the claim being tested, so it is out.
Step 3 — Check the direction: The organization thinks the mean is now higher, which points right. Option b points left, leaving option d.
$$ H_{0}: \mu = 4.5 \qquad H_{a}: \mu > 4.5 $$Answer: d. \(H_{0}: \mu = 4.5\), \(H_{a}: \mu > 4.5\)
Key Terms
null hypothesis — written \(H_{0}\), the statement of no difference or no effect; the status quo claim we assume while examining the evidence.
alternative hypothesis — written \(H_{a}\), the claim contradicting \(H_{0}\); what we conclude when the evidence is strong enough to reject the null.
reject \(H_{0}\) — the decision made when the sample information favors the alternative hypothesis.
do not reject \(H_{0}\) — the decision made when the sample information is insufficient to reject the null hypothesis; it is not a claim that the null is true.