Question

(16.19) A class survey in a large class for first-year college students asked, About how many hours do you study during a typical week? The mean response of the 463 students was x = 15.3 hours. Suppose that we know that the study time follows a Normal distribution with standard deviation σ = 8.5 hours in the population of all first-year students at this university.

Step 1: Use the survey result to give a 99% confidence interval for the mean study time of all first-year students.

A) 24.08 to 28.92

B) 14.98 to 21.02

C) 14.28 to 16.32

D) 17.64 to 18.36

Step 2: What condition not yet mentioned is needed for your confidence interval to be valid?

A) Normal distribution

B) SRS

C) Known σ

D) Unknown μ

Answer 1

From the given information, the mean response of the 463 students was I = 15.3 hours and the study time follows a Normal distribution with standard deviation o=8.5 hours in the population of all first-year students at this university. That is, T = 15.3,0 = 8.5 and n= 463. Step 1: The objective is to construct a 99% confidence interval for the mean study time of all first- year students. The level of significance is, a=1-0.99 a=0.01 Compute the 99% confidence interval. CI = 7+Zaza = 15.3 +2.58 (From z distribution table, 20.012= 20.005 = 2.58) =(15.3-1.0192,15.3+1.0192)

The z table is given as follows: Standard Normal Distribution Table (Right-Tail Probabilities) 2 .00 0.0 -5000 .01 4960 .02 4920 .03 4880 .04 4840 .05 .06 .07 4801 4761 4721 .08 .09 4681 4641 2.4 2.2 | .0139 .0136 .0132 .0129 .0125 .0122.0119.0116 .0113 .0110 2.3.0107 .0104 .0102.0099 .0096 .0094 .0091 .0089.0087 .0084 .0082.0080.0078 .0075 .0073 .0071 .0069.0068 .0066.0064 2.5 | .0062.0060.0059.0057.0055 .0054 .0052.0051 .0049 .0048 2.6 .0047 .0045 .0044.0043 .0041 .0040.0039.0038 0037.0036 2.7.0035 .0034 .0033 .0032 .0031 .0030 .0029.0028 .0027 .0026 2.8 .0026 .0025 .0024.0023 .0023 .0022 .0021 .0021.0020 .0019 Therefore, option C is correct. Step 2: The condition of randomization is not yet mentioned. The data must be sampled randomly. The condition of SRS (simple random sample) is needed for the confidence interval to be valid, which is not yet mentioned. Therefore, option B is correct.