A sample of 51 night-school students' ages is obtained in order to estimate the mean age of night-school students. x = 26 years. The population variance is 19.
(a) Give a point estimate for μ. (Give your answer correct to one decimal place.)
(b) Find the 95% confidence interval for μ. (Give your answer correct to two decimal places.)
Lower Limit
Upper Limit
(c) Find the 99% confidence interval for μ. (Give your answer correct to two decimal places.)
Lower Limit
Upper Limit
95% = mean ± 1.96 SEm
99% = mean ± 2.575 SEm
SEm = SD/√n
SD^2 = variance
I'll let you do the calculations.
(a) The point estimate for μ is the sample mean. Therefore, the point estimate is x = 26 years.
(b) To find the 95% confidence interval, we can use the formula:
CI = x ± (Z * (σ/√n))
Where,
CI is the confidence interval,
x is the sample mean,
Z is the critical value (for a 95% confidence level, Z = 1.96),
σ is the population standard deviation, and
n is the sample size.
Plug in the given values:
x = 26 years,
Z = 1.96,
σ = √19 ≈ 4.36,
n = 51.
CI = 26 ± (1.96 * (4.36 / √51))
Calculate:
CI = 26 ± (1.96 * 0.611)
CI = 26 ± 1.196
Lower Limit = 26 - 1.196 = 24.804
Upper Limit = 26 + 1.196 = 27.196
Therefore, the 95% confidence interval for μ is (24.80, 27.20).
(c) To find the 99% confidence interval, we can use the same formula as above, but with a different critical value.
CI = x ± (Z * (σ / √n))
For a 99% confidence level, Z = 2.57.
Plug in the given values:
x = 26 years,
Z = 2.57,
σ = √19 ≈ 4.36,
n = 51.
CI = 26 ± (2.57 * (4.36 / √51))
Calculate:
CI = 26 ± (2.57 * 0.611)
CI = 26 ± 1.570
Lower Limit = 26 - 1.570 = 24.430
Upper Limit = 26 + 1.570 = 27.570
Therefore, the 99% confidence interval for μ is (24.43, 27.57).
To solve these questions, we need to use the formula for confidence intervals, which is given by:
Confidence Interval = x ± z * (σ / √n)
Where:
- x is the sample mean,
- z is the z-score (also known as the critical value),
- σ is the population standard deviation,
- n is the sample size.
Now let's solve each part of the question step by step:
(a) Point estimate for μ:
The point estimate for μ is simply the sample mean x, which is given as 26 years.
(b) 95% confidence interval for μ:
For a 95% confidence interval, we need to find the z-score corresponding to a confidence level of 95%. By referring to the Z-table or using a calculator, we find that the z-score for a 95% confidence level is 1.96.
Substituting the values into the formula:
Confidence Interval = 26 ± 1.96 * (√19 / √51)
Confidence Interval = 26 ± 1.96 * (4.358898944 / 7.141428429)
Confidence Interval = 26 ± 1.96 * 0.610829
Calculating the confidence interval:
Lower Limit = 26 - (1.96 * 0.610829)
Lower Limit = 26 - 1.197151
Lower Limit = 24.802849
Upper Limit = 26 + (1.96 * 0.610829)
Upper Limit = 26 + 1.197151
Upper Limit = 27.197151
So the 95% confidence interval for μ is (24.80, 27.20).
(c) 99% confidence interval for μ:
Similarly, for a 99% confidence interval, we need to find the z-score corresponding to a confidence level of 99%. By referring to the Z-table or using a calculator, we find that the z-score for a 99% confidence level is 2.58.
Substituting the values into the formula:
Confidence Interval = 26 ± 2.58 * (√19 / √51)
Confidence Interval = 26 ± 2.58 * (4.358898944 / 7.141428429)
Confidence Interval = 26 ± 2.58 * 0.610829
Calculating the confidence interval:
Lower Limit = 26 - (2.58 * 0.610829)
Lower Limit = 26 - 1.573436
Lower Limit = 24.426564
Upper Limit = 26 + (2.58 * 0.610829)
Upper Limit = 26 + 1.573436
Upper Limit = 27.573436
So the 99% confidence interval for μ is (24.43, 27.57).