Question

Consider the following random sample of diameter measurements (in inches) of 14 softballs: 4.86, 4.89, 4.71, 4.87, 4.74, 4.77, 4.75, 4.7, 4.69, 4.85, 4.68, 4.69, 4.88, 4.79 If we assume that the diameter measurements are normally distributed, find a 95% confidence interval for the mean diameter of a softball. Then complete the table below. Carry your intermediate computations to at least three decimal places. Round your answers to two decimal places. (If necessary, consult a list of formulas.) What is the lower limit of the confidence interval? What is the upper limit of the confidence interval?

Answer 1

Solution:

x	x2
4.86	23.6196
4.89	23.9121
4.71	22.1841
4.87	23.7169
4.74	22.4676
4.77	22.7529
4.75	22.5625
4.7	22.09
4.69	21.9961
4.85	23.5225
4.68	21.9024
4.69	21.9961
4.88	23.8144
4.79	22.9441
∑x=66.87	∑x2=319.4813

Mean ˉx=∑xn

=4.86+4.89+4.71+4.87+4.74+4.77+4.75+4.7+4.69+4.85+4.68+4.69+4.88+4.79/14

=66.87/14

=4.7764

Sample Standard deviation S=√∑x2-(∑x)2nn-1

=√319.4813-(66.87)214/13

=√319.4813-319.399813

=√0.0815/13

=√0.0063

=0.079

Degrees of freedom = df = n - 1 = 14 - 1 = 13

At 95% confidence level the t is ,

  = 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

t /2,df = t0.025,13 =2.797  

Margin of error = E = t/2,df * (s /n)

= 2.797 * (0.08 / 14)

= 0.046

Margin of error = 0.046

The 95% confidence interval estimate of the population mean is,

- E < < + E

4.78 - 0.046 < < 4.78 + 0.046

4.734 < < 4.826

The lower limit =4.734

Tthe upper limit = 4.826