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 2 comma 816 adults in this sample yielded statistics on network​ size, that​ is, the mean number of people named per person was x overbarequals14.8​, with a standard deviation of sequals9.9.

Complete parts a through d.

a. Give a point estimate for mu.

b.Give an interval estimate for μ. Use a confidence coefficient of 0.95

c- 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​.

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?

Answer 1

a) The point estimate of is same as the sample mean. Hence point estimate of = 14.8.

b) Since the population variance is unknown hence we would be using the t-statistics to estimate the population mean with a confidence coefficient of 0.95.

The sample size is 2816 which is very high. Degree of freedom = 2816 - 1 = 2815.

Remember that at such a high sample size value, the critical value of the t statistics will become approximately equal to the critical value of the z statistics.

For a confidence coefficient of 0.95, t critical is

Hence, the interval estimate of is given by

c) The statement is correct. 95% confidence interval means that in 95% of the cases the true population means will lie in the interval calculated in part b.

d) The shape of the distribution doesn't impact how well the sample mean matches the population mean. In reality, any data will seldom be perfectly normally distributed. But whatever be the shape of the distribution, the 95% confidence interval will always be valid. Even if the distribution is skewed, in 95% of the cases, the true population means will lie in the interval calculated in part b.

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