Question

 

1. Confidence Interval: You collect a random sample of size N and compute ?̅=50. The sample standard deviation (s) equals 10. Suppose the true mean μ is equal to 45. Compute a 95% confidence interval for the below sample sizes N. Which of the 95% confidence intervals contain the true mean (μ) and which do not? Explain.

a) N=16

b) N=36

c) N=100

2. Sampling Distribution: A population of FIU students has an average height of 68 inches with a standard deviation of 4 inches.

a) What is the probability that a randomly selected student has a height of 6 feet or taller?

b) What is the probability that the average height of 9 randomly selected students has a height of 6 feet or taller?

Answer 1

1)a) n = 16

     df = 16 - 1 15

At 95% confidence interval the critical value is t^* = 2.131

The 95% confidence interval for population mean is

+/- t^* * s/

= 50 +/- 2.131 * 10/

= 50 +/- 5.3275

= 44.6725, 55.3275

The true mean value 45 lies in the confidence interval.

b) n = 36

At 95% confidence interval the critical value is z^* = 1.96

The 95% confidence interval for population mean is

+/- z^* * s/

= 50 +/- 1.96 * 10/

= 50 +/- 3.27

= 46.73, 53.27

The true mean value 45 doesn't lie in the confidence interval.

c) n = 100

At 95% confidence interval the critical value is z^* = 1.96

The 95% confidence interval for population mean is

+/- z^* * s/

= 50 +/- 1.96 * 10/

= 50 +/- 1.96

= 48.04, 51.96

The true mean value 45 doesn't lie in the confidence interval.

2) P(X > 72)

= P((X - )/> (72 - )/)

= P(Z > (72 - 68)/4)

= P(Z > 1)

= 1 - P(Z < 1)

= 1 - 0.8413

= 0.1587

b) P((> 72)

= P(( - )/() > (72 - )/())

= P(Z > (72 - 68)/(4/))

= P(Z > 3)

= 1 - P(Z < 3)

= 1 - 0.9987

= 0.0013