Question

Here are summary statistics for randomly selected weights of newborn girls: n = 163, x = 28.6 hg, S=6.9 hg. Construct a confidence interval estimate of the mean. Use a 95% confidence level. Are these results very different from the confidence interval 27.6 hg < <30.0 hg with only 20 sample values, x = 28.8 hg, and s = 2.5 hg? What is the confidence interval for the population mean u? ng <u< hg (Round to one decimal place as needed.) Are the results between the two confidence intervals very different? O A. No, because the confidence interval limits are similar. OB. No, because each confidence interval contains the mean of the other confidence interval. O C. Yes, because the confidence interval limits are not similar. OD. Yes, because one confidence interval does not contain the mean of the other confidence interval.

Answer 1

SOLUTION:

From given data,

Here are summary statistics for randomly selected weights of newborn​ girls: n=163​, $\bar{x}$ =28.6 hg, s = 6.9 hg. Construct a confidence interval estimate of the mean.

Use a 95​% confidence level. Are these results very different from the confidence interval 27.6 hg < $\mu$ < 30.0 hg with only 20 sample values, $\bar{x}$ = 28.8 hg, and s = 2.5 ​hg?

n = Sample Size = 163

$\bar{x}$ =Sample Mean = 28.6 hg

s = Sample standard deviation = 6.9 hg

What is the confidence interval for the population mean $\mu$ ?

SE = s / sqrt(n)

SE = 6.9 / sqrt(163) =0.540449

Take

95% confidence interval

95/100 = 0.95

Significance level = $\alpha$ = 1-0.95 = 0.05

$\alpha$/2 = 0.05/2 = 0.025

Degree of freedom = df = n - 1 = 163 - 1 = 162

From Table,

critical values of t

t_{$\alpha$/2,df} = t_0.025,162 = 1.9747

Confidence Interval:

$\bar{x}$$\pm$ t_{$\alpha$/2,df} * SE

28.6 $\pm$ (1.9747 * 0.540449)

28.6 $\pm$ 1.0672246403

( 28.6 - 1.0672246403 , 28.6+1.0672246403)

(27.6 , 29.7)

27.6 hg <  $\mu$ <  29.7 hg (Rounded to one decimal place.)

Are the results between the two confidence intervals very different?

We have the confidence interval 27.6 hg < $\mu$ < 30.0 hg.

Answer : option (A)

No, because the confidence interval limits are similar.

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