Question

The random sample shown below was selected from a normal distribution.

3, 5, 9, 4, 6, 9

Complete parts a and b.

A. Construct a 95% confidence interval for the population mean μ .

(___,___)

B. Assume that sample mean x and sample standard deviation s remain exactly the same as those you just calculated but that are based on a sample of n = 25 observations. Repeat part a. What is the effect of increasing the sample size on the width of the confidence intervals

The confidence interval is (___,___)

What is the effect of the sample size on the width of the confidence interval

A. As the sample size​ increases, the width decreases.

B. As the sample size​ increases, the width increases.

C. As the sample size​ increases, the width stays the same.

A. Many ancient tombs were cut from limestone rock that contained uranium. Since most such tombs are not​ well-ventilated, they may contain radon gas. In one​ study, the radon levels in a sample of 12 tombs in a particular region were measured in becquerels per cubic meter (Bq/m3). For this​ data, assume that x(over-bar)= 3,611 Bq/m3 and s= 1,269 Bq/m3. Use this information to​ estimate, with 95% confidence, the mean level of radon exposure in tombs in the region. Interpret the resulting interval.

The confidence interval is (_,_)

Answer 1

A.

Sample mean, = (3 + 5 + 9 + 4 + 6 + 9)/6 = 6

Sample Variance = [(3 - 6)^2 + (5 - 6)^2 + (9 - 6)^2 + (4 - 6)^2 + (6 - 6)^2 + (9 - 6)^2 ]/(6 - 1) = 6.4

Sample Standard deviation, s = = 2.53

Standard error of mean , SE = 2.53 / = 1.033

Degree of freedom, df = n-1 = 6-1 = 5

Critical value of t at df = 5 and  95% confidence interval is 2.57

95% confidence interval for the population mean μ is,

(6 - 2.57 * 1.033 ,  6 + 2.57 * 1.033)

(3.345,  8.655)

Confidence interval length = 8.655 - 3.345 = 5.31

B.

Standard error of mean , SE = 2.53 /   = 0.506

Degree of freedom, df = n-1 = 25-1 = 24

Critical value of t at df = 24 and  95% confidence interval is 2.06

95% confidence interval for the population mean μ is,

(6 - 2.06 * 0.506 ,  6 + 2.06 * 0.506)

(4.958,  7.042)

Confidence interval length = 7.042 - 4.958 = 2.084

A. As the sample size​ increases, the width decreases.

A.

Standard error of mean , SE = 1269 /   = 366.3287

Degree of freedom, df = n-1 = 12-1 = 11

Critical value of t at df = 11 and  95% confidence interval is 2.20

95% confidence interval for the population mean μ is,

(3611 - 2.20 * 366.3287 , 3611 + 2.20 * 366.3287)

(2805.077,  4416.923)

There is a 95% chance that the confidence interval (2805.077,  4416.923) contains the true population mean radon levels measured in becquerels per cubic meter.