Question

A government official is in charge of allocating social programs throughout the city of Vancouver. He will decide where these social outreach programs should be located based on the percentage of residents living below the poverty line in each region of the city. He takes a simple random sample of 120 people living in Gastown and finds that 23 have an annual income that is below the poverty line. For each of the following statements, specify whether the statement is a correct interpretation of the 95% confidence interval for the true proportion of Gastown residents living below the poverty line. A. 19.17%(23/120) of Gastown residents are living below the poverty line. B. There is a 95% probability that the true proportion of Gastown residents who are living below the poverty line equals 23/120. C. If another random sample of 120 Gastown residents is drawn, there is a 95% probability that the sample proportion of Gastown residents who are living below the poverty line equals 23/120. D. If many random samples of 120 Gastown residents are drawn, 95% of the resulting confidence intervals will contain the value of the true proportion of Gastown residents who are living below the poverty line. E. If many random samples of 120 Gastown residents are drawn, 95% of the resulting confidence intervals will contain the value 23/120. Note: You can earn partial credit on this problem.

Answer 1

The correct interpretations of the 95% confidence interval for the true proportion of Gastown residents living below the poverty line are explained.

Step-by-step explanation:

The correct interpretation of the 95% confidence interval for the true proportion of Gastown residents living below the poverty line are as follows:

A. 19.17%(23/120) of Gastown residents are living below the poverty line is not a correct interpretation. The 95% confidence interval gives us a range of values within which the true proportion is likely to fall, but it does not tell us the exact proportion.

B. There is not a 95% probability that the true proportion of Gastown residents who are living below the poverty line equals 23/120. The confidence interval provides information about the range of likely values, but it does not assign probabilities to specific values.

C. If another random sample of 120 Gastown residents is drawn, there is not a 95% probability that the sample proportion of Gastown residents who are living below the poverty line equals 23/120. The confidence interval is specific to the sample at hand and does not carry over to future samples.

D. If many random samples of 120 Gastown residents are drawn, 95% of the resulting confidence intervals will contain the value of the true proportion of Gastown residents who are living below the poverty line. This is a correct interpretation. The confidence interval gives us a range of likely values, and in 95% of the samples, the true proportion will fall within that range.

E. If many random samples of 120 Gastown residents are drawn, 95% of the resulting confidence intervals will contain the value 23/120. This is not a correct interpretation. The confidence interval provides a range of likely values for the proportion, not a specific value.