Question

You are looking at a population and are interested in the proportion p that has a certain characteristic. Unknown to you, this population proportion is p=0.60. You have taken a random sample of size n=120 from the population and found that the proportion of the sample that has the characteristic is p ^ ​ =0.53. Your sample is Sample 1 in the table below. (In the table, Sample 1 is written "S1", Sample 2 is written "S2", etc.) (a) Based on Sample 1, graph the 75% and 95% for the population proportion. Use 1.150 for the critical value for the 75% confidence interval, and use 1.960 for the critical value for the 95% confidence interval. (If necessary, consult a - Enter the lower and upper limits on the graphs to show each confidence interval. Write your answers with two decimal places. - For the points ( ⋆ and ⋆ ), enter the population proportion, 0.60. (b) Press the "Generate Samples" button below to simulate taking 19 more samples of size n=120 from the same population. Notice that the confidence - matically. Then complete parts (c) and (d) below the table. (c) Notice that for 20 18 ​ =90% of the samples, the 95% confidence interval contains the population proportion. Choose the correct statement. When constructing 95% confidence intervals for 20 samples of the same size from the population, exactly 95% of the samples must contain the population proportion. There must have been an error with the way our samples were chosen. When constructing 95% confidence intervals for 20 samples of the same size from the population, the percentage of the samples that contain the population proportion should be close to 95%, but it may not be exactly 95%. When constructing 95% confidence intervals for 20 samples of the same size from the population, at most 95% of the samples will contain the population proportion. (d) Choose ALL that are true. To guarantee that a confidence interval will contain the population proportion, the level of confidence must match the sample proportion. For example, if p ​ equals 0.45, then the 45% confidence interval will contain the population proportion. From the 75% confidence interval for Sample 6, we cannot say that there is a 75% probability that the population proportion is between 0.39 and 0.78. The 75% confidence interval for Sample 6 is narrower than the 95% confidence interval for Sample 6 . This must be the case; when constructing a confidence interval for a sample, the greater the level of confidence, the wider the confidence interval. If there were a Sample 21 of size n=160 with the same sample proportion as Sample 6 , then the 75% confidence interval for Sample 21 would be wider than the 75% confidence interval for Sample 6. None of the choices above are true.

Answer 1

(a) The 75% confidence interval for p is [0.54, 0.66], and the 95% confidence interval is [0.51, 0.69].

(c) When constructing 95% CIs for 20 samples, the percentage containing p may be close to 95%, but not exactly.

(d) True: Confidence level must match sample proportion for CI guarantee. If Sample 21 had the same proportion as Sample 6, its 75% CI would be wider.

Step-by-step explanation:

(a) For the 75% confidence interval, we use the critical value 1.150. The margin of error (ME) is calculated as ME = critical value * standard error. For a 75% confidence interval, ME = 1.150 * sqrt(0.60 * (1 - 0.60) / 120) ≈ 0.06. Therefore, the interval is [0.53 - 0.06, 0.53 + 0.06], which simplifies to [0.54, 0.66].

For the 95% confidence interval, with a critical value of 1.960, ME = 1.960 * sqrt(0.60 * (1 - 0.60) / 120) ≈ 0.08. The interval is [0.53 - 0.08, 0.53 + 0.08], which simplifies to [0.51, 0.69].

(c) The statement that best fits the scenario is: When constructing 95% confidence intervals for 20 samples of the same size from the population, the percentage of the samples that contain the population proportion should be close to 95%, but it may not be exactly 95%.

(d) The correct statements are: To guarantee that a confidence interval will contain the population proportion, the level of confidence must match the sample proportion. If there were a Sample 21 of size n=160 with the same sample proportion as Sample 6, then the 75% confidence interval for Sample 21 would be wider than the 75% confidence interval for Sample 6.