Question

A research program used a representative random sample of men and women to gauge the size of the personal network of older adults. Each adult in the sample was asked to​ "please name the people you have frequent contact with and who are also important to​ you." The responses of 2824 adults in this sample yielded statistics on network​ size, that​ is, the mean number of people named per person was x=14.6, with a standard deviation of s=10.3 . Complete parts a through d.a- Give a point estimate for μ.b- Give an interval estimate for μ. Use a confidence coefficient of 0.95c- Comment on the validity of the following​statement: "95% of the​ time, the true mean number of people named per person will fall in the interval computed in part b​."Choose the correct answer below.A. The statement is correct.​ 95% of the​ time, the true mean number of people named per person will fall within an interval computed with a confidence coefficient of 0.95.B. The statement is incorrect. A correct statement would be​"One can be​ 95% confident that the true mean number of people named per person will fall in the interval computed in part b.​"C. The statement is incorrect. A correct statement would be​"95% of the​ time, the true mean number of people named per person will fall outside the interval computed in part b.D. The statement is incorrect. A correct statement would be​"One can be​ 95% confident that the true mean number of people named per person will fall outside the interval computed in part b.​d- It is unlikely that the personal network sizes of adults are normally distributed. In​ fact, it is likely that the distribution is highly skewed. If​ so, what​ impact, if​ any, does this have on the validity of inferences derived from the confidence​interval?A. It does impact the validity of the interpretation because the interpretation is based on highly skewed resultsB. It does impact the validity of the interpretation because the interpretation was based upon a sample instead of the entire population.C. It does not impact the validity of the interpretation because the interpretation is based on highly skewed results.D. It does not impact the validity of the interpretation because the sampling space of the sample mean is approximately normal according to the Central Limit Theorem.

Answer 1

Answer:

a. $\mu=\bar x =14.6$

b. The 95% CI for the population mean is (14.22, 14.98).

c. B. "The statement is incorrect. A correct statement would be​"One can be​ 95% confident that the true mean number of people named per person will fall in the interval computed in part b"

d. D. It does not impact the validity of the interpretation because the sampling space of the sample mean is approximately normal according to the Central Limit Theorem.

Step-by-step explanation:

a) The sample mean provides a point estimation of the population mean.

In this case, the estimation of the mean is:

$\mu=\bar x =14.6$

b) With the information of the sample we can estimate the

As the sample size n=2824 is big enough, we can aproximate the t-statistic with a z-statistic.

For a 95% CI, the z-value is z=1.96.

The sample standard deviation is s=10.3.

The margin of error of the confidence is then calculated as:

$E=z\cdot s/\sqrt{n}=1.96*10.3/\sqrt{2824}=20.188/53.141=0.38$

The lower and upper limits of the CI are:

$LL=\bar x-z\cdot s/\sqrt{n}=14.6-0.38=14.22\\\\UL=\bar x+z\cdot s/\sqrt{n}=14.6+0.38=14.98$

The 95% CI for the population mean is (14.22, 14.98).

c. "95% of the​ time, the true mean number of people named per person will fall in the interval computed in part b"

The right answer is:

B. "The statement is incorrect. A correct statement would be​"One can be​ 95% confident that the true mean number of people named per person will fall in the interval computed in part b"

The confidence interval gives bounds within there is certain degree of confidence that the true population mean will fall within.

It does not infer nothing about the sample means or the sampling distribution. It only takes information from a sample to estimate a interval for the population mean with certain degree of confidence.

d. It is unlikely that the personal network sizes of adults are normally distributed. In​ fact, it is likely that the distribution is highly skewed. If​ so, what​ impact, if​ any, does this have on the validity of inferences derived from the confidence​ interval?

The answer is:

D. It does not impact the validity of the interpretation because the sampling space of the sample mean is approximately normal according to the Central Limit Theorem.

The reliability of a confidence interval depends more on the sample size, not on the distribution of the population. As the sample size increases, the absolute value of the skewness and kurtosis of the sampling distribution decreases. This sample size relationship is expressed in the central limit theorem.