Question

Find and interpret a \(95 \%\) confidence interval for a population mean \(\mu\) for these values: a. \(n=36, \bar{x}=13.1, s^{2}=3.42\) b. \(n=64, \bar{x}=2.73, s^{2}=.1047\)

Short answer

Answer: For the first set of values, the 95% confidence interval for the population mean μ is between 12.49 and 13.71. For the second set of values, the 95% confidence interval for the population mean μ is between 2.65 and 2.81. These intervals can be interpreted to mean that we are 95% confident that the true population mean falls within these ranges for each respective set of values.

Step-by-step solution

Identify the given values

In this exercise, we are given two different sets of values (a and b), which include the sample size (\(n\)), sample mean (\(\bar{x}\)), and the sample variance (\(s^2\)). First, let's identify the given values for each case: a. \(n=36, \bar{x}=13.1, s^{2}=3.42\) b. \(n=64, \bar{x}=2.73, s^{2}=.1047\)

Calculate the standard deviation

Now, calculate the sample standard deviation (\(s\)) by taking the square root of the sample variance (\(s^2\)) for each case: a. \(s = \sqrt{3.42} \approx 1.85\) b. \(s = \sqrt{.1047} \approx 0.32\)

Find the standard error

The standard error (\(SE\)) represents the uncertainty in the sample mean estimation. It can be calculated by dividing the standard deviation by the square root of the sample size: \(SE = \frac{s}{\sqrt{n}}\) Calculate the standard error for each case: a. \(SE_a = \frac{1.85}{\sqrt{36}} \approx 0.31\) b. \(SE_b = \frac{0.32}{\sqrt{64}} \approx 0.04\)

Determine the Confidence Interval using the z score

For a 95% confidence interval, the z score for a standard normal distribution is approximately 1.96. We can find the margin of error (\(ME\)) and compute the confidence interval using the formulas: \(ME = 1.96 \times SE\) \(CI = (\bar{x} - ME, \bar{x} + ME)\) Calculate the confidence interval for each case: a. \(ME_a = 1.96 \times 0.31 \approx 0.61\) \(CI_a = (13.1 - 0.61, 13.1 + 0.61) = (12.49, 13.71)\) b. \(ME_b = 1.96 \times 0.04 \approx 0.08\) \(CI_b = (2.73 - 0.08, 2.73 + 0.08) = (2.65, 2.81)\)

Interpret the Confidence Intervals

Now that we have the 95% confidence intervals, we can interpret them as follows: a. We are 95% confident that the population mean \(\mu\) is between 12.49 and 13.71 for the first set of values. b. We are 95% confident that the population mean \(\mu\) is between 2.65 and 2.81 for the second set of values.