Answer :
a) The 95% confidence interval is (137.6, 148.4). b) The 95% confidence interval is (227.3, 237.1). c) The 95% confidence interval is (58.1, 60.1).
a. To calculate the 95% confidence interval estimate for the population mean in part (a), we can use the following formula:
(X - 1.96σ/√n, X + 1.96σ/√n)
where:
X is the sample mean
σ is the population standard deviation (we are assuming that we know the population standard deviation in this case)
n is the sample size
In this case, we have:
X = 143
σ = 9
n = 70
Plugging these values into the formula, we get the following confidence interval:
(143 - 1.96 × 9/√70, 143 + 1.96 × 9/√70)
(137.6, 148.4)
Therefore, we can be 95% confident that the true population mean is between 137.6 and 148.4.
b. To calculate the 95% confidence interval estimate for the population mean in part (b), we can use the following formula:
(X - 1.96σ/√n, X + 1.96σ/√n)
where:
X is the sample mean
σ is the sample standard deviation
n is the sample size
In this case, we have:
X = 232.2
σ = 9
n = 80
Plugging these values into the formula, we get the following confidence interval:
(232.2 - 1.96 × 9/√80, 232.2 + 1.96 × 9/√80)
(227.3, 237.1)
Therefore, we can be 95% confident that the true population mean is between 227.3 and 237.1.
c. To calculate the 95% confidence interval estimate for the population mean in part (c), we can use the following formula:
(X - 1.96√[tex]\sigma^2[/tex]/√n, X + 1.96√[tex]\sigma^2[/tex]/√n)
where:
X is the sample mean
σ^2 is the population variance (we are not given the population standard deviation in this case, so we have to estimate it using the sample standard deviation s)
n is the sample size
In this case, we have:
X = 59.1
[tex]s^2[/tex] = 126
n = 200
We can estimate the population variance [tex]\sigma^2[/tex] as follows:
[tex]\sigma^2[/tex] = [tex]s^2[/tex]/n = 126/200 = 0.63
Plugging these values into the formula, we get the following confidence interval:
(59.1 - 1.96√0.63/√200, 59.1 + 1.96√0.63/√200)
(58.1, 60.1)
Therefore, we can be 95% confident that the true population mean is between 58.1 and 60.1.
To learn more about confidence interval here:
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