A random sample of size 81 has sample mean 20 and sample standard deviation 3.
(a) Is it appropriate to use a Student’s t distribution to compute a confidence interval for the population mean μ? Explain.
(b) Find a 95% confidence interval for μ.
(c) Explain the meaning of the confidence interval you computed.
(a) To determine if it is appropriate to use a Student's t distribution to compute a confidence interval for the population mean μ, we need to consider a few factors. The Student's t distribution is commonly used when the population standard deviation is unknown and the sample size is small (typically less than 30). In this case, we have a sample size of 81, which is larger than 30. Additionally, we have the sample standard deviation, which can provide a good estimate for the population standard deviation. Hence, with a large sample size and the availability of the sample standard deviation, it is more appropriate to use a normal distribution in this scenario rather than a Student's t distribution.
(b) To find a 95% confidence interval for μ, we can use the formula:
Confidence Interval = sample mean ± (critical value * sample standard deviation / sqrt(sample size))
In this case, we have:
Sample mean (x̄) = 20
Sample standard deviation (s) = 3
Sample size (n) = 81
To determine the critical value, we need to use the appropriate z-value for a 95% confidence level. For a two-tailed test, the critical z-value is approximately 1.96.
Plugging in the values into the formula, we get:
Confidence Interval = 20 ± (1.96 * 3 / sqrt(81))
Simplifying the expression further:
Confidence Interval = 20 ± (1.96 * 3 / 9)
Confidence Interval = 20 ± 0.65
Therefore, the 95% confidence interval for μ is (19.35, 20.65).
(c) The confidence interval we computed represents the range of values within which we can be 95% confident that the population mean μ lies. In this case, we are 95% confident that the true population mean μ falls between 19.35 and 20.65. This means that if we were to take multiple random samples and construct confidence intervals using the same method, approximately 95% of these intervals would contain the true population mean.