Suppose that a random sample of five observations was taken from a normal population whose
variance is 25. The results are 8, 15, 12, 6, 7. Find the 99% confidence interval estimate of the
population mean.
To find the 99% confidence interval estimate of the population mean, we need to follow these steps:
Step 1: Calculate the sample mean.
Given the sample observations: 8, 15, 12, 6, 7
The sample mean (x̄) can be calculated by summing up all the observations and dividing it by the sample size:
x̄ = (8 + 15 + 12 + 6 + 7) / 5 = 48 / 5 = 9.6
Step 2: Calculate the sample standard deviation.
The sample standard deviation (s) can be calculated using the formula:
s = √((Σ(x - x̄)^2) / (n - 1))
where Σ denotes the sum, x is each individual observation, x̄ is the sample mean, and n is the sample size.
Using the given observations, we can calculate the sum of squared differences from the mean:
(8 - 9.6)^2 + (15 - 9.6)^2 + (12 - 9.6)^2 + (6 - 9.6)^2 + (7 - 9.6)^2 = 291.2
Now, divide this sum by (n - 1) to get the sample variance:
s^2 = 291.2 / (5 - 1) = 291.2 / 4 = 72.8
Finally, take the square root of the sample variance to get the sample standard deviation:
s = √(72.8) ≈ 8.54
Step 3: Determine the critical value.
Since the population standard deviation is unknown, we use the t-distribution instead of the standard normal distribution. For a 99% confidence interval with (n - 1) degrees of freedom (where n is the sample size), we will use the t-distribution table to find the critical value. In this case, with n = 5-1 = 4 degrees of freedom, the critical value is approximately 3.747.
Step 4: Calculate the margin of error.
The margin of error (E) can be calculated using the formula:
E = t * (s / √n)
where t is the critical value, s is the sample standard deviation, and n is the sample size.
Substituting the values:
E = 3.747 * (8.54 / √5) ≈ 7.64
Step 5: Calculate the confidence interval.
The confidence interval can be calculated as:
CI = (x̄ - E, x̄ + E)
Substituting the values:
CI = (9.6 - 7.64, 9.6 + 7.64)
CI ≈ (1.96, 17.24)
Therefore, the 99% confidence interval estimate of the population mean is approximately (1.96, 17.24).