Question

Use the pulse rates for a random sample of adult women listed below to determine a 90% confidence interval estimate for the population average pulse rate for all adult women. Assume that pulse rate is normally distributed.

83 58 70 56 76 64 80 76 70 97 68 78 108

Use the data above, find an 80% confidence interval estimate for the population variance in pulse rate.

Answer 1

Answer 2

To determine a 90% confidence interval estimate for the population average pulse rate for all adult women, we can use the formula for the confidence interval of the mean:

Confidence Interval = x̄ ± Z * (σ / √n),

where: x̄ is the sample mean, Z is the Z-score corresponding to the desired confidence level (90% in this case), σ is the population standard deviation (unknown), n is the sample size.

First, let's calculate the sample mean (x̄) and sample standard deviation (s) using the given data:

Sample mean (x̄) = (83 + 58 + 70 + 56 + 76 + 64 + 80 + 76 + 70 + 97 + 68 + 78 + 108) / 13 = 77.6154

To calculate the sample standard deviation, we need to find the variance first:

Sample variance (s^2) = Σ((xi - x̄)^2) / (n - 1) = [(83 - 77.6154)^2 + (58 - 77.6154)^2 + ... + (108 - 77.6154)^2] / (13 - 1) = 5613.9231

Sample standard deviation (s) = √(s^2) = √(5613.9231) = 74.9223

Next, we need to find the Z-score corresponding to a 90% confidence level. Using a standard normal distribution table or a calculator, we find that the Z-score for a 90% confidence level is approximately 1.645.

Now we can calculate the confidence interval:

Confidence Interval = x̄ ± Z * (σ / √n) = 77.6154 ± 1.645 * (74.9223 / √13) = 77.6154 ± 20.4867

Therefore, the 90% confidence interval estimate for the population average pulse rate for all adult women is (57.1287, 98.1021).

To find an 80% confidence interval estimate for the population variance in pulse rate, we can use the chi-square distribution.

The formula for the confidence interval of the variance is:

Confidence Interval = [(n - 1) * s^2] / χ^2,

where: n is the sample size, s^2 is the sample variance, χ^2 is the chi-square value corresponding to the desired confidence level (80% in this case) and (n - 1) degrees of freedom.

The degrees of freedom for variance estimation is (n - 1).

First, let's calculate the chi-square value corresponding to an 80% confidence level and (n - 1) degrees of freedom. Using a chi-square distribution table or a calculator, we find that the chi-square value is approximately 17.328.

Now we can calculate the confidence interval:

Confidence Interval = [(n - 1) * s^2] / χ^2 = [(13 - 1) * 5613.9231] / 17.328 = 1984.7439

Therefore, the 80% confidence interval estimate for the population variance in pulse rate is (1984.7439, ∞).