Question

8. Suppose that (unknown to you) 78% of all undergraduates favor eliminating supplemental fees for lab courses. If you took a very large number of simple random samples of size
   n = 100 from this population, the sampling distribution of the sample proportion would be approximately Normal with a mean of 0.78 and a standard deviation of

   1. 0.100.

   2. 0.002.

   3. 0.410.

   4. 0.041.

   5. There is not enough information available to compute the standard deviation.

9. A surprising fact is that only 48% of adults use a password on their mobile device. If a poll chooses a simple random sample of 500 adults and asks if they use a password on their mobile device, the percentage who say “Yes” will naturally vary from sample to sample, if the sampling is repeated. In fact, if we look at the sampling distribution, we will see that the percentage who say “Yes” in many samples will follow a Normal distribution, with a meanof 48% and a standard deviation of 1.2%. In other words, our sampling distribution will be centered at 48% and will have a standard deviation of 1.2%. From this information, we know the middle 68% of the sampling distribution will be between what two values?

   A. 47.5% and 48.5% B. 46.8% and 49.2% C. 45.6% and 50.4% D. 46% and 50%

   E. 48% and 100%

10. You and a friend want to estimate the proportion of undergraduates at OSU who favor eliminating classes that begin before 9 a.m. Your friend will choose a simple random sample of 900 students from all undergraduate students who are currently attending OSU. You plan to choose a simple random sample of 90 students from all undergraduate students who are currently attending OSU. Think carefully about what you’d expect the sampling distributionof the sample proportion to look like for samples of size 900 versus samples of size
    90. When compared to the sampling distribution based on samples of size 900, the sampling distribution based on samples of size 90 will have

    A. a much smaller mean (or center).
    B. a much larger mean (or center).
    C. approximately the same mean (or center).
    D. It’s impossible to answer this question without more information.

Answer 1

Solution:
Question 1)

p = proportions of all undergraduates favor eliminating supplemental fees for lab courses = 0.78
n = sample size = 100

the sampling distribution of the sample proportion would be approximately Normal with a mean of 0.78 and

a standard deviation of :

Thus correct option is: 0.041

Question 2)

Given: the percentage who say “Yes” in many samples will follow a Normal distribution, with a mean of 48% and a standard deviation of 1.2%.

the middle 68% of the sampling distribution will be between what two values?

According to empirical rule for bell shaped distribution ( Normal distribution )

68% of the data falls within 1 standard deviation from mean
that is :

Thus find:

and

Thus the middle 68% of the sampling distribution will be between B. 46.8% and 49.2%

Question 3)

You and a friend want to estimate the proportion of undergraduates at OSU who favor eliminating classes that begin before 9 a.m.

Your friend sample size = n = 900 students

Your sample size = n = 90

When compared to the sampling distribution based on samples of size 900, the sampling distribution based on samples of size 90 will have :

C. approximately the same mean (or center).

Since sample proportion is an unbiased estimator of population proportion and here both sample sizes are large. So we can say: both have approximately same mean.