Question

1. When σ is unknown and the sample is of size n ≥ 30, there are two methods for computing confidence intervals for μ.

Method 1: Use the Student's t distribution with d.f. = n − 1.
This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method.

Method 2: When n ≥ 30, use the sample standard deviation s as an estimate for σ, and then use the standard normal distribution.
This method is based on the fact that for large samples, s is a fairly good approximation for σ. Also, for large n, the critical values for the Student's t distribution approach those of the standard normal distribution.

Consider a random sample of size n = 36, with sample mean x = 45.1 and sample standard deviation s = 6.0.

(a) Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

	90%	95%	99%
lower limit
upper limit

(b) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

	90%	95%	99%
lower limit
upper limit

(d) Now consider a sample size of 81. Compute 90%, 95%, and 99% confidence intervals for μ using Method 1 with a Student's t distribution. Round endpoints to two digits after the decimal.

	90%	95%	99%
lower limit
upper limit

(e) Compute 90%, 95%, and 99% confidence intervals for μ using Method 2 with the standard normal distribution. Use s as an estimate for σ. Round endpoints to two digits after the decimal.

	90%	95%	99%
lower limit
upper limit

2. The home run percentage is the number of home runs per 100 times at bat. A random sample of 43 professional baseball players gave the following data for home run percentages.

1.6	2.4	1.2	6.6	2.3	0.0	1.8	2.5	6.5	1.8
2.7	2.0	1.9	1.3	2.7	1.7	1.3	2.1	2.8	1.4
3.8	2.1	3.4	1.3	1.5	2.9	2.6	0.0	4.1	2.9
1.9	2.4	0.0	1.8	3.1	3.8	3.2	1.6	4.2	0.0
1.2	1.8	2.4

(a) Use a calculator with mean and standard deviation keys to find x and s. (Round your answers to two decimal places.)

x =	%
s =	%

(b) Compute a 90% confidence interval for the population mean μ of home run percentages for all professional baseball players. Hint: If you use the Student's t distribution table, be sure to use the closest d.f. that is smaller. (Round your answers to two decimal places.)

lower limit	%
upper limit	%

(c) Compute a 99% confidence interval for the population mean μ of home run percentages for all professional baseball players. (Round your answers to two decimal places.)

lower limit	%
upper limit	%

Answer 1

Solution1A

In excel use confidence.t to get the confidence intervals using t

and confidence .norm to get the confidence intervals using z.

for example for 90%

margin of error=CONFIDENCE.T(0.1;6;36)

=1.689572

using method 2

margin of error=CONFIDENCE.NORM(0.1;6;36)=1.959964

lower limit=sample mean-margin of error

and sample mean+margin of errror

here sample mean=45.1

	n=36	using method1
xbar=45.1	margin of error	lower limit	upper limit
90%	1.689572	43.41	46.79
95%	2.030108	43.07	47.13
99%	2.723806	42.38	47.82
	n=36	using method2
xbar=45.1	margin of error	lower limit	upper limit
90%	1.644854	43.46	46.74
95%	1.959964	43.14	47.06
99%	2.575829	42.52	47.68
	n=81	using method1
xbar=45.1	margin of error	lower limit	upper limit
90%	1.109416	43.99	46.21
95%	1.326709	43.77	46.43
99%	1.759127	43.34	46.86
	n=81	using method2
xbar=45.1	margin of error	lower limit	upper limit
90%	1.096569	44.00	46.20
95%	1.306643	43.79	46.41
99%	1.71722	43.38	46.82