Assume pulse rate is normally distributed. Use the pulse rates for a random samp

Question

Assume pulse rate is normally distributed. Use the pulse rates for a random sample of adult women listed below to determine a 95% confidence interval estimate for the population mean in pulse rate for all adult women.

83  70 56 76 64 80 76 70 97 68 78 108

3. Use the data in question #2 to find an 80% confidence interval estimate for the population variance in pulse rates of adult women.

4.    a. If we increase the confidence level in #2 to 99%, in general what happens to the width of the confidence interval?

b. If there were 36 subjects in the study (instead of 12) but we keep the confidence level at 99% (part a), what happens to the width of the confidence interval?

Explanation / Answer

Confidence interval

2. Assume pulse rate is normally distributed. Use the pulse rates for a random sample of adult women listed below to determine a 95% confidence interval estimate for the population mean in pulse rate for all adult women.

83 70 56 76 64 80 76 70 97 68 78 108

Solution:

The required confidence interval for the population mean is given as below:

Confidence Interval Estimate for the Mean

Data

Sample Standard Deviation

14.13463402

Sample Mean

77.16666667

Sample Size

12

Confidence Level

95%

Intermediate Calculations

Standard Error of the Mean

4.080317377

Degrees of Freedom

11

t Value

2.2010

Interval Half Width

8.9807

Confidence Interval

Interval Lower Limit

68.19

Interval Upper Limit

86.15

3. Use the data in question #2 to find an 80% confidence interval estimate for the population variance in pulse rates of adult women.

Solution:

The required confidence interval for the population variance is given as below:

Confidence Interval Estimate for the Population Variance

Data

Sample Size

12

Sample Standard Deviation

14.13463

Confidence Level

80%

Intermediate Calculations

Degrees of Freedom

11

Sum of Squares

2197.665418

Single Tail Area

0.1

Lower Chi-Square Value

5.5778

Upper Chi-Square Value

17.2750

Results

Interval Lower Limit for Variance

127.2165

Interval Upper Limit for Variance

394.0033

Interval Lower Limit for Standard Deviation

11.2790

Interval Upper Limit for Standard Deviation

19.8495

Assumption:

Population from which sample was drawn has an approximate normal distribution.

4.    a. If we increase the confidence level in #2 to 99%, in general what happens to the width of the confidence interval?

Solution:

The 99% confidence interval is given as below:

Confidence Interval Estimate for the Mean

Data

Sample Standard Deviation

14.13463402

Sample Mean

77.16666667

Sample Size

12

Confidence Level

99%

Intermediate Calculations

Standard Error of the Mean

4.080317377

Degrees of Freedom

11

t Value

3.1058

Interval Half Width

12.6727

Confidence Interval

Interval Lower Limit

64.49

Interval Upper Limit

89.84

The 99% confidence interval = (64.49, 89.84)

The 95% Confidence interval = (68.19, 86.15)

It is observed that as we increase the confidence level, the width of the confidence interval is increases.

b. If there were 36 subjects in the study (instead of 12) but we keep the confidence level at 99% (part a), what happens to the width of the confidence interval?

Solution:

Confidence Interval Estimate for the Mean

Data

Sample Standard Deviation

14.13463402

Sample Mean

77.16666667

Sample Size

36

Confidence Level

99%

Intermediate Calculations

Standard Error of the Mean

2.355772337

Degrees of Freedom

35

t Value

2.7238

Interval Half Width

6.4167

Confidence Interval

Interval Lower Limit

70.75

Interval Upper Limit

83.58

The 99% confidence interval with sample size 36 = (70.75, 83.58)

The 99% confidence interval with sample size 12 = (64.49, 89.84)

This means, as we keep confidence level constant, and if we increase the sample size then the width of the confidence interval is decreases.

Confidence Interval Estimate for the Mean

Data

Sample Standard Deviation

14.13463402

Sample Mean

77.16666667

Sample Size

12

Confidence Level

95%

Intermediate Calculations

Standard Error of the Mean

4.080317377

Degrees of Freedom

11

t Value

2.2010

Interval Half Width

8.9807

Confidence Interval

Interval Lower Limit

68.19

Interval Upper Limit

86.15

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