A study considered the question, "Are you a registered voter?" Accuracy of response was confirmed by a check of city voting records. Two methods of survey were used: a face-to-face interview and a telephone interview. A random sample of 84 people was asked the voter registration question face-to-face. Seventy respondents gave accurate answers (as verified by city records). Another random sample of 74 people was asked the same question during a telephone interview. Sixty-five respondents gave accurate answers. Assume that the samples are representative of the general population. Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some cases, this choice will increase the P-value by a small amount or increase the length of the confidence interval, thereby making the answer slightly more "conservative."(a) Let p1 be the population proportion of all people who answer the voter registration question accurately during a face-to-face interview. Let p2 be the population proportion of all people who answer the question accurately during a telephone interview. Find a 95% confidence interval for p1 − p2. (Round your answers to four decimal places.)lower limit upper limit (b) Does the interval contain numbers that are all positive? all negative? mixed? Comment on the meaning of the confidence interval in the context of this problem. At the 95% level, do you detect any difference in the proportion of accurate responses from face-to-face interviews compared with the proportion of accurate responses from telephone interviews?Since the interval contains positive and negative values, we can say, at the 95% level, that there is no significant difference between the proportion of accurate responses for face-to-face interviews and that for telephone interviews.Since the interval contains only negative values, we can say, at the 95% level, that the proportion of accurate responses for face-to-face interviews is less than that for telephone interviews. Since the interval contains only positive values, we can say, at the 95% level, that the proportion of accurate responses for face-to-face interviews is greater than that for telephone interviews. (c) Test the claim that there is a difference in the proportion of accurate responses from face-to-face interviews compared with telephone interviews. Use α = 0.05.(i) What is the level of significance? State the null and alternate hypotheses.H0: p1 > p2; H1: p1 < p2H0: p1 < p2; H1: p1 > p2 H0: p1 = p2; H1: p1 ≠ p2H0: p1 ≠ p2; H1: p1 = p2H0: p1 = p2; H1: p1 < p2 (ii) What sampling distribution will you use? What assumptions are you making?We will use Student's t distribution. We assume that both population distributions are approximately normal and that σ1 and σ2 are unknown.We will use Student's t distribution. We assume that d has a normal distribution and that σ1 and σ2 are unknown. We will use the standard normal distribution. We assume that both population distributions are approximately normal and that σ1 and σ2 are known.We will use the standard normal distribution. We assume the number of trials must be sufficiently large.We will use the standard normal distribution. We assume that d has a normal distribution and that σ1 and σ2 are known. What is the value of the sample test statistic? (Round your answer to two decimal places.) (iii) Find the P-value. (Round your answer to four decimal places.) Sketch the sampling distribution and show the area corresponding to the P-value. (iv) Based on your answers in parts (i) to (iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?Since the P-value > 0.05, we fail to reject H0. The data are not statistically significant.Since the P-value ≤ 0.05, we fail to reject H0. The data are statistically significant. Since the P-value > 0.05, we reject H0. The data are not statistically significant.Since the P-value ≤ 0.05, we reject H0. The data are statistically significant. (v) Interpret your conclusion in the context of the application.Reject H0. At the 5% level of significance, there is insufficient evidence to conclude that the population proportion of adults who give accurate answers differs between face-to-face and telephone interviews.Fail to reject H0. At the 5% level of significance, there is sufficient evidence to conclude that the population proportion of adults who give accurate answers differs between face-to-face and telephone interviews. Reject H0. At the 5% level of significance, there is sufficient evidence to conclude that the population proportion of adults who give accurate answers differs between face-to-face and telephone interviews.Fail to reject H0. At the 5% level of significance, there is insufficient evidence to conclude that the population proportion of adults who give accurate answers differs between face-to-face and telephone interviews.
A study considered the question, "Are you a registered voter?" Accuracy of response was confirmed by a check of city voting records. Two methods of survey were used: a face-to-face interview and a telephone interview. A random sample of 84 people was asked the voter registration question face-to-face. Seventy respondents gave accurate answers (as verified by city records). Another random sample of 74 people was asked the same question during a telephone interview. Sixty-five respondents gave accurate answers. Assume that the samples are representative of the general population. Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some cases, this choice will increase the P-value by a small amount or increase the length of the confidence interval, thereby making the answer slightly more "conservative."
(a) Let p1 be the population proportion of all people who answer the voter registration question accurately during a face-to-face interview. Let p2 be the population proportion of all people who answer the question accurately during a telephone interview. Find a 95% confidence interval for p1 − p2. (Round your answers to four decimal places.)
lower limit
upper limit
(b) Does the interval contain numbers that are all positive? all negative? mixed? Comment on the meaning of the confidence interval in the context of this problem. At the 95% level, do you detect any difference in the proportion of accurate responses from face-to-face interviews compared with the proportion of accurate responses from telephone interviews?
Since the interval contains positive and negative values, we can say, at the 95% level, that there is no significant difference between the proportion of accurate responses for face-to-face interviews and that for telephone interviews.
Since the interval contains only negative values, we can say, at the 95% level, that the proportion of accurate responses for face-to-face interviews is less than that for telephone interviews.
Since the interval contains only positive values, we can say, at the 95% level, that the proportion of accurate responses for face-to-face interviews is greater than that for telephone interviews.
(c) Test the claim that there is a difference in the proportion of accurate responses from face-to-face interviews compared with telephone interviews. Use α = 0.05.
(i) What is the level of significance?
State the null and alternate hypotheses.
H0: p1 > p2; H1: p1 < p2
H0: p1 < p2; H1: p1 > p2
H0: p1 = p2; H1: p1 ≠ p2
H0: p1 ≠ p2; H1: p1 = p2
H0: p1 = p2; H1: p1 < p2
(ii) What sampling distribution will you use? What assumptions are you making?
We will use Student's t distribution. We assume that both population distributions are approximately normal and that σ1 and σ2 are unknown.
We will use Student's t distribution. We assume that d has a normal distribution and that σ1 and σ2 are unknown.
We will use the standard normal distribution. We assume that both population distributions are approximately normal and that σ1 and σ2 are known.
We will use the standard normal distribution. We assume the number of trials must be sufficiently large.
We will use the standard normal distribution. We assume that d has a normal distribution and that σ1 and σ2 are known.
What is the value of the sample test statistic? (Round your answer to two decimal places.)
(iii) Find the P-value. (Round your answer to four decimal places.)
Sketch the sampling distribution and show the area corresponding to the P-value.
(iv) Based on your answers in parts (i) to (iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?
Since the P-value > 0.05, we fail to reject H0. The data are not statistically significant.
Since the P-value ≤ 0.05, we fail to reject H0. The data are statistically significant.
Since the P-value > 0.05, we reject H0. The data are not statistically significant.
Since the P-value ≤ 0.05, we reject H0. The data are statistically significant.
(v) Interpret your conclusion in the context of the application.
Reject H0. At the 5% level of significance, there is insufficient evidence to conclude that the population proportion of adults who give accurate answers differs between face-to-face and telephone interviews.
Fail to reject H0. At the 5% level of significance, there is sufficient evidence to conclude that the population proportion of adults who give accurate answers differs between face-to-face and telephone interviews.
Reject H0. At the 5% level of significance, there is sufficient evidence to conclude that the population proportion of adults who give accurate answers differs between face-to-face and telephone interviews.
Fail to reject H0. At the 5% level of significance, there is insufficient evidence to conclude that the population proportion of adults who give accurate answers differs between face-to-face and telephone interviews.
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