5. 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. (1 mark)
(b) Find a 95% confidence interval for μ. (3 marks)
(c) Explain the meaning of the confidence interval you computed. (1 mark)
(a) To determine whether it is appropriate to use a Student's t distribution, we need to consider two conditions:
1. Random Sampling: It is mentioned that the sample is "random," which means the observations were selected independently and randomly from the population. This condition is met in the given scenario since the sample is stated to be random.
2. Sample Size: The sample size must be sufficiently large for the t-distribution to approximate the standard normal distribution. A commonly used guideline is that the sample size should be greater than or equal to 30. In this case, we have a sample size of 81, which meets this condition.
Therefore, both conditions are satisfied, and it is appropriate to use a Student's t distribution.
(b) To find a 95% confidence interval for μ, we can use the formula:
Confidence interval = sample mean ± (t-multiplier * standard error)
Given:
Sample mean (x̄) = 20
Sample standard deviation (s) = 3
Sample size (n) = 81
First, we need to calculate the standard error, which is the standard deviation of the sample mean:
Standard error (SE) = s / √n
SE = 3 / √81
SE = 3 / 9
SE = 1/3
Next, we need to find the t-multiplier for a 95% confidence interval with 80 degrees of freedom (n-1 = 81-1 = 80). We can refer to the t-distribution table or use statistical software to find this value. For a 95% confidence level with 80 degrees of freedom, the t-multiplier is approximately 1.994.
Now we can calculate the confidence interval:
Confidence interval = 20 ± (1.994 * (1/3))
Confidence interval = 20 ± 0.665
Therefore, the 95% confidence interval for μ is (19.335, 20.665).
(c) The meaning of the confidence interval is that we can be 95% confident that the true population mean (μ) lies within the calculated interval (19.335, 20.665). This means if we were to repeat the sampling process many times and calculate the confidence intervals, approximately 95% of those intervals would contain the true population mean.