Statistics questions

MTH 245 Lesson 19 Notes One-Sample Proportion Test

To perform a one-sample proportion hypothesis test using StatCrunch:

1. Open a blank StatCrunch data table. 2. Click on Stat  Proportion Stats  One Sample  With Summary.

3. Enter the number of successes and the number of observations.

4. Leave the radio button at "Hypothesis test for p" (the default).

5. Fill in the null hypothesis value and the alternative hypothesis operator.

6. Click "Compute!". Example 1: The manufacturer of a certain anti-cholesterol medication claims that among patients who take the drug, the chances they will develop headaches as a side effect is at most four percent. In a random sample of 94 patients prescribed the medication, six reported experiencing headaches as a side effect. At significance level 𝛼𝛼 = 0.01, is there enough evidence to reject the company's claim?

Identify the correct null and alternative hypotheses.

𝐻𝐻0: 𝑝𝑝 = 0.04 (Original claim: 𝑝𝑝 ≤ 0.04)

𝐻𝐻𝐴𝐴: 𝑝𝑝 > 0.04

What is the p-value? (Round to three decimal places as needed.) 0.119

State and interpret the appropriate decision for this hypothesis test.

Since the p-value = 0.119 > 𝛼𝛼 = 0.01, we fail to reject 𝐻𝐻0. There is insufficient evidence to reject the manufacturer's claim that at most four percent of patients who take the drug will develop headaches as a side effect.

Example 2: A researcher claims that 86% of college graduates say their college degree has been a good investment. In a random sample of 1,000 graduates, 845 reported that they consider their college degree to be a good investment. At significance level 𝛼𝛼 = 0.10, is there enough evidence to reject the researcher's claim?

Identify the correct null and alternative hypotheses.

𝐻𝐻0: 𝑝𝑝 = 0.86 (Original claim: 𝑝𝑝 = 0.86)

𝐻𝐻𝐴𝐴: 𝑝𝑝 ≠ 0.86

What is the p-value? (Round to three decimal places as needed.) 0.172

State and interpret the appropriate decision for this hypothesis test.

Since the p-value = 0.172 > 𝛼𝛼 = 0.10, we fail to reject 𝐻𝐻0. There is insufficient evidence to reject the researcher's claim that 86% of college graduates say their college degree has been a good investment.

Example 3: A polling organization claims that at most 75% of U. S. adults think that drivers are safer using hands-free devices instead of hand-held cell phones. In a random sample of 150 U.S. adults, 77% agreed with the statement "drivers are safer using hands-free devices." At significance level 𝛼𝛼 = 0.01, is there enough evidence to reject the organization's claim?

Identify the correct null and alternative hypotheses.

𝐻𝐻0: 𝑝𝑝 = 0.75 (Original claim: 𝑝𝑝 ≤ 0.75)

𝐻𝐻𝐴𝐴: 𝑝𝑝 > 0.75

What is the p-value? (Round to three decimal places as needed.)

Note: in this case, we are not given the actual count of individuals in the sample of 150 who agreed with the statement, only a percentage. To get the raw count, we multiply the number in the sample by the percentage who agreed: 150 × 0.77 = 115.5. Because this is a count, not a measurement, we round to the nearest integer to get 116, then input that number into StatCrunch to get a p-value of 0.255.

State and interpret the appropriate decision for this hypothesis test.

Since the p-value = 0.255 > 𝛼𝛼 = 0.01, we fail to reject 𝐻𝐻0. There is insufficient evidence to reject the manufacturer's claim that organization claims that at most 75% of U. S. adults think that drivers are safer using hands-free devices instead of hand-held cell phones.

Example 4: A humane society claims that fewer than 35% of U. S. households own at least one cat. In a random sample of 400 U. S. households, 126 are found to own at least one cat. At significance level 𝛼𝛼 = 0.10, is there enough evidence to support the society's claim?

Identify the correct null and alternative hypotheses.

𝐻𝐻0: 𝑝𝑝 = 0.35

𝐻𝐻𝐴𝐴: 𝑝𝑝 < 0.35 (Original claim: 𝑝𝑝 < 0.35)

What is the p-value? (Round to three decimal places as needed.) 0.071

State and interpret the appropriate decision for this hypothesis test.

Since p-value = 0.071 < 𝛼𝛼 = 0.10, we reject 𝐻𝐻0. There is sufficient evidence to support the society's claim that that fewer than 35% of U. S. households own at least one cat.