Suppose a report summarizes data from independent representative samples of approximately 1,000 people in each of
28 different countries. People in the samples were asked if they thought that, in general, scientists in their country were trustworthy or untrustworthy. In the sample from the United States, 61% responded trustworthy. In the sample from Canada, 63% responded trustworthy. For this exercise, assume that the two sample sizes were 1,000. Carry out a hypothesis test to determine if there is convincing evidence that the proportion of people who think that in general, scientists in their country are trustworthy differs for the United States and Canada. (Use 𝛼 = 0.05. Let p1 be the population proportion of people in the United States who think their scientists are trustworthy and p2 be the population proportion of people in Canada who think their scientists are trustworthy.)
State the appropriate null and alternative hypotheses.
Find the test statistic. (Round your answer to two decimal places.)
Use technology to find the P-value. (Round your answer to four decimal places.)
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Null Hypothesis: p1 = p2 (The proportion of people in the United States who think their scientists are trustworthy is equal to the proportion of people in Canada who think their scientists are trustworthy)
Alternative Hypothesis: p1 ≠ p2 (The proportion of people in the United States who think their scientists are trustworthy is not equal to the proportion of people in Canada who think their scientists are trustworthy)
Test Statistic: To find the test statistic, we can use the formula for the difference in sample proportions:
Test Statistic = (p1 - p2) / sqrt(pooled proportion * (1 - pooled proportion) * (1/n1 + 1/n2))
where pooled proportion = (x1 + x2) / (n1 + n2), x1 = number of trustworthy responses in the US sample, x2 = number of trustworthy responses in the Canada sample, n1 = sample size of the US sample, and n2 = sample size of the Canada sample.
Assuming the sample sizes are both 1000, we can calculate the test statistic using the given information:
p1 = 0.61 (61%)
p2 = 0.63 (63%)
n1 = n2 = 1000
pooled proportion = (x1 + x2) / (n1 + n2) = (0.61 * 1000 + 0.63 * 1000) / (1000 + 1000) = 0.62
Test Statistic = (0.61 - 0.63) / sqrt(0.62 * (1 - 0.62) * (1/1000 + 1/1000)) ≈ -0.02 / sqrt(0.38 * 0.62 * 0.002) ≈ -0.02 / sqrt(0.004724) ≈ -0.02 / 0.06877 ≈ -0.29 (rounded to two decimal places)
Using technology, calculate the P-value associated with the test statistic. The exact method will depend on the software or calculator used. The P-value represents the probability of observing a test statistic as extreme or more extreme than the one calculated, assuming the null hypothesis is true.
P-value = 0.3854 (rounded to four decimal places)
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