MS Module 4 t values and confidence intervals practice problems

(The attached PDF file has better formatting.)

Exercise 4.1: t values and confidence intervals

A statistician estimates the population mean for a normal distribution from a sample of 8 points. The 99% confidence interval for the population mean is (0, 2.500)

A.    What is the critical t value for a 99% confidence interval from a sample of 8 points?
B.    What is the standard deviation of the sample?
C.    What is the critical t value for a 95% confidence interval from a sample of 8 points?
D.    What is the 95% confidence interval for the population mean?

Part A: A sample with 8 points has 7 degrees of freedom. The critical t value for a 99% confidence interval with 7 degrees of freedom is 3.499 (table lookup).

Part B: The confidence interval is the estimate ± σ/√N × t value, so

●    the point estimate is (2.500 + 0) / 2 = 1.250
●    the width of the confidence interval is 2 × σ/√N × t value = 2.500 – 0 = 2.500 ➾
    ○    σ = 80.5 × 2.500 / (2 × 3.499) = 1.010

Part C: The critical t value for a 95% confidence interval with 7 degrees of freedom is 2.365 (table lookup).

Part D: The confidence interval is 1.250 ± (1.010 / 80.5) × 2.365:

●    lower bound: 1.250 – (1.010 / 80.5) × 2.365 = 0.405
●    upper bound: 1.250 + (1.010 / 80.5) × 2.365 = 2.095



Exercise 4.2: t values and two confidence intervals

A statistician estimates the population mean for a normal distribution from a sample of 8 points.

●    The upper bound of the 95% confidence interval for the population mean is 5.
●    The lower bound of the 90% confidence interval for the population mean is 1.

We use the following notation:

●     μ = the estimated population mean
●     σ = the standard deviation of the sample
●     N = the number of observations

A.    What is the critical t value for a two-sided 95% confidence interval for a sample of 8 points?
B.    What is the critical t value for a two-sided 90% confidence interval for a sample of 8 points?
C.    What is σ/√N?
D.    What is σ?
E.    What is μ?

Part A: A sample of 8 points has 8 – 1 = 7 degrees of freedom.

The critical t value for a two-sided 95% confidence interval for a sample of 8 points is 2.3646 (table look-up).

Part B: A sample of 8 points has 8 – 1 = 7 degrees of freedom.

The critical t value for a two-sided 90% confidence interval for a sample of 8 points is 1.8946 (table look-up).

Part C: Combine the upper half of the 95% confidence interval with the lower half of the 90% confidence interval to estimate σ/√N:

●    2.3646 × σ/√N = 5 – μ
●    1.8946 × σ/√N = μ – 1

adding: (2.3646 + 1.8946) × σ/√N = (5 – μ) + (μ – 1) = (5 – 1) ➾

σ/√N = (5 – 1) / (2.3646 + 1.8946) = 0.93914

Part D: σ = 0.93914 × 80.5 = 2.656

Part E: Solve for μ by either of the confidence intervals:

●    2.3646 × σ/√N = 5 – μ ➾ μ = 5 – 2.3646 × 0.93914 = 2.779
●    1.8946 × σ/√N = μ – 1 ➾ μ = 1 + 1.8946 × 0.93914 = 2.779