Step-by-Step Guidance for Statistics Hypothesis Testing and Confidence Intervals

Q1. Compute the value of the test statistic for the claim that μ > 28, given n = 50, x̄ = 28.3, σ = 1.2, at α = 0.05.

Background

Topic: Hypothesis Testing for a Population Mean (Z-test)

This question tests your ability to calculate the test statistic for a one-sided hypothesis test about a population mean when the population standard deviation is known.

Key Terms and Formulas

• μ: Population mean

• x̄: Sample mean

• σ: Population standard deviation

• n: Sample size

• Z-test statistic formula:

$Z = \frac{x\bar - \mu_0}{\frac{\sigma}{\sqrt{n}}}$

Step-by-Step Guidance

1. Identify the null hypothesis value: $\mu_0 = 28$.

2. Calculate the standard error: $SE = \frac{\sigma}{\sqrt{n}} = \frac{1.2}{\sqrt{50}}$.

3. Subtract the hypothesized mean from the sample mean: $x\bar - \mu_0 = 28.3 - 28$.

4. Set up the Z-test statistic formula: $Z = \frac{28.3 - 28}{SE}$.

Try solving on your own before revealing the answer!

Q2. Find the 95.44% confidence interval for the mean forced vital capacity for all asthmatics, given n = 15, σ = 0.6, and the sample data provided.

Background

Topic: Confidence Interval for a Population Mean (Normal Distribution, σ Known)

This question tests your ability to construct a confidence interval for the mean when the population standard deviation is known and the sample size is small.

Key Terms and Formulas

• Confidence interval formula:

$CI = x\bar \pm z^* \frac{\sigma}{\sqrt{n}}$

• $z^*$: Critical value for the desired confidence level (for 95.44%, $z^* \approx 2$)

• $x\bar$: Sample mean (calculate from the data)

• $\sigma$: Population standard deviation

• $n$: Sample size

Step-by-Step Guidance

1. Calculate the sample mean $x\bar$ by adding all the data values and dividing by 15.

2. Find the critical value $z^*$ for a 95.44% confidence interval (look up or recall the value).

3. Compute the standard error: $SE = \frac{0.6}{\sqrt{15}}$.

4. Set up the confidence interval formula: $CI = x\bar \pm z^* \times SE$.

Try solving on your own before revealing the answer!

Q3. Test the claim that μ = 18.7 months for convicted burglars using the critical value method, given n = 11, x̄ = 21.3, s = 7.7, α = 0.05.

Background

Topic: Hypothesis Testing for a Population Mean (t-test, Small Sample, σ Unknown)

This question tests your ability to perform a hypothesis test for a population mean using the t-distribution when the population standard deviation is unknown and the sample size is small.

Key Terms and Formulas

• Null hypothesis: $H_0: \mu = 18.7$

• Alternative hypothesis: $H_1: \mu \neq 18.7$

• Test statistic formula:

$t = \frac{x\bar - \mu_0}{\frac{s}{\sqrt{n}}}$

• Degrees of freedom: $df = n - 1$

• Critical value: Find $t_{\alpha/2}$ for $df = 10$

Step-by-Step Guidance

1. State the null and alternative hypotheses.

2. Calculate the standard error: $SE = \frac{7.7}{\sqrt{11}}$.

3. Compute the t-test statistic: $t = \frac{21.3 - 18.7}{SE}$.

4. Find the critical value $t_{\alpha/2}$ for $df = 10$ at $\alpha = 0.05$.

Try solving on your own before revealing the answer!

Q4. Test the claim that the population mean weight of employees is less than 200 lb, given n = 54, x̄ = 183.9 lb, σ = 121.2 lb, α = 0.10.

Background

Topic: One-Sided Hypothesis Test for a Population Mean (Z-test, σ Known)

This question tests your ability to perform a one-sided hypothesis test for a population mean using the Z-test when the population standard deviation is known.

Key Terms and Formulas

• Null hypothesis: $H_0: \mu = 200$

• Alternative hypothesis: $H_1: \mu < 200$

• Z-test statistic formula:

$Z = \frac{x\bar - \mu_0}{\frac{\sigma}{\sqrt{n}}}$

Step-by-Step Guidance

1. State the null and alternative hypotheses.

2. Calculate the standard error: $SE = \frac{121.2}{\sqrt{54}}$.

3. Compute the Z-test statistic: $Z = \frac{183.9 - 200}{SE}$.

4. Find the critical value $z_{\alpha}$ for a left-tailed test at $\alpha = 0.10$.

Try solving on your own before revealing the answer!