following data values are a simple random sample from a population that is normally distributed, with ssquared=25.0: 47, 43, 33, 42, 34, and 41. Construct and interpret the 95% and 99% confidence intervals for the population mean.
To construct confidence intervals for the population mean, we can use the formula:
Confidence Interval = sample mean ± (critical value * standard error)
First, we need to calculate the sample mean and standard error.
Sample Mean (x̄):
x̄ = (47 + 43 + 33 + 42 + 34 + 41) / 6
x̄ = 240 / 6
x̄ = 40
Standard Error (SE):
SE = √(ssquared / n)
SE = √(25.0 / 6)
SE ≈ 2.89
Next, we need to determine the critical values for the confidence intervals.
For a 95% confidence interval, we need to find the critical value for a two-tailed test with alpha = 0.05. Since the sample size is small (n < 30), we can use the t-distribution instead of the z-distribution.
Degrees of Freedom (df):
df = n - 1
df = 6 - 1
df = 5
Critical Value (CV) for 95% confidence interval:
CV = t(0.025, 5) = -2.571
For a 99% confidence interval, the critical value for a two-tailed test with alpha = 0.01 is:
CV = t(0.005, 5) = -4.032
Now we can calculate the confidence intervals.
95% Confidence Interval:
Lower Limit = x̄ - (CV * SE)
Lower Limit = 40 - (-2.571 * 2.89)
Lower Limit ≈ 47.45
Upper Limit = x̄ + (CV * SE)
Upper Limit = 40 + (-2.571 * 2.89)
Upper Limit ≈ 32.55
The 95% confidence interval for the population mean is approximately (32.55, 47.45). This means that we are 95% confident that the true population mean falls within this interval.
99% Confidence Interval:
Lower Limit = x̄ - (CV * SE)
Lower Limit = 40 - (-4.032 * 2.89)
Lower Limit ≈ 53.82
Upper Limit = x̄ + (CV * SE)
Upper Limit = 40 + (-4.032 * 2.89)
Upper Limit ≈ 26.18
The 99% confidence interval for the population mean is approximately (26.18, 53.82). This means that we are 99% confident that the true population mean falls within this interval.
Interpretation: In the 95% confidence interval, we are 95% confident that the true population mean lies between 32.55 and 47.45. In the 99% confidence interval, we are 99% confident that the true population mean lies between 26.18 and 53.82.