Question

Among various ethnic groups, the standard deviation of heights is known to be approximately three inches. We wish

to construct a 95% confidence interval for the mean height of male Swedes. Forty-eight male Swedes are surveyed. The

sample mean is 71 inches. The sample standard deviation is 2.8 inches.

a.

I. $\overline{X}$=________

ii. σ =________

iii. n =________

b. In words, define the random variables X and X

c. Which distribution should you use for this problem? Explain your choice.

d. Construct a 95% confidence interval for the population mean height of male Swedes.

I. State the confidence interval.

ii. Sketch the graph.

iii. Calculate the error bound.

e. What will happen to the level of confidence obtained if 1,000 male Swedes are surveyed instead of 48? Why?

Short answer

a.(I) $\overline{\overline{\overline{X}}}$= 71inches

(ii) $\sigma =3$inches

(iii) n = 48

b. The average height of 48 Swiss guys is $\overline{x}$, and the Swiss male's height is X.

c. The sample size is greater than 30.

d. (I) CI - (70.151,71,49)

(ii)

(iii) EBM = 0.849

e. we do not require a larger interval to capture the true population mean.

Step-by-step solution

Explanation of A

I. The average height of 48 male Swedes in the study was 71 inches.

$\overline{x}=71$inches

ii. The population of distinct ethnic groups has a 3 inch standard variance in height.

$\sigma =3$inches

iii. The total number of male Swedes polled from various ethnic groups is, n=48

Explanation of B

The random variables X and $\overline{X}$:

The average height of 48 Swiss guys is role="math" localid="1648640680776" $\overline{X}$, and the Swiss male's height is X.

Explanation of C

The problem is solved with a normal distribution. For the population's standard deviation, we know that the sample size is greater than 30.

Explanation of D

I. State the confidence interval. CI - (70.151,71,49)

ii. The graph is given below,

iii. The error bound is calculated from the formula,

EBM =$\frac{upperlimit-lowerlimit}{2}$

EBM = $\frac{71.849–70.151}{2}$

EBM = 0.849

Explanation of E

The confidence interval will shrink in size as the sample size grows. As we know, when all parameters remain constant, increasing the sample size reduces variability.

As a result, we do not require a larger interval to capture the true population mean.