Step-by-Step Guidance for College Statistics Exam Practice

Q1. Confidence Intervals for a Proportion

Background

Topic: Estimating population proportions and constructing confidence intervals.

This question tests your ability to calculate point estimates, check requirements, and build a confidence interval for a proportion.

Key Terms and Formulas:

• Point estimate for proportion: $\hat{p} = \frac{x}{n}$

• Complement: $\hat{q} = 1 - \hat{p}$

• Confidence interval formula: $\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}\hat{q}}{n}}$

Step-by-Step Guidance

1. Calculate the sample proportion: $\hat{p} = \frac{440}{800}$

2. Find $\hat{q}$: $\hat{q} = 1 - \hat{p}$

3. Check if $n\hat{p} \geq 5$ and $n\hat{q} \geq 5$ to verify requirements for a confidence interval.

4. Set up the confidence interval formula for a 95% confidence level: $\hat{p} \pm z_{0.025} \sqrt{\frac{\hat{p}\hat{q}}{n}}$ (where $z_{0.025} = 1.96$).

Try solving on your own before revealing the answer!

Final Answer:

$\hat{p} = 0.55$, $\hat{q} = 0.45$

95% confidence interval: $0.55 \pm 1.96 \sqrt{\frac{0.55 \times 0.45}{800}}$

The interval does support the claim that more than half use tablets, since the lower bound is above 0.5.

Q2. Determining Sample Size for a Proportion

Background

Topic: Sample size calculation for estimating proportions with a specified margin of error and confidence level.

This question tests your ability to use formulas to determine the minimum sample size needed for a desired accuracy.

Key Terms and Formulas:

• Sample size formula (no prior estimate): $n = \frac{z_{\alpha/2}^2 \cdot 0.25}{E^2}$

• Sample size formula (with prior estimate): $n = \frac{z_{\alpha/2}^2 \cdot \hat{p} \hat{q}}{E^2}$

• Margin of error: $E = 0.02$

• Confidence level: 99% ($z_{\alpha/2} = 2.576$)

Step-by-Step Guidance

1. For no prior estimate, use $\hat{p} = 0.5$ and $\hat{q} = 0.5$ in the formula.

2. Plug in $z_{\alpha/2} = 2.576$ and $E = 0.02$ into the formula.

3. For the prior estimate ($\hat{p} = 0.65$, $\hat{q} = 0.35$), substitute these values into the formula.

4. Compare the sample sizes and consider how the value of $\hat{p}$ affects the required sample size.

Try solving on your own before revealing the answer!

Final Answer:

No prior estimate: $n = \frac{(2.576)^2 \times 0.25}{(0.02)^2} \approx 4156$

With prior estimate: $n = \frac{(2.576)^2 \times 0.65 \times 0.35}{(0.02)^2} \approx 3792$

Having a prior estimate closer to 0 or 1 decreases the required sample size.

Q3. Inference for a Single Mean (t-distribution)

Background

Topic: Constructing confidence intervals for a population mean using the t-distribution.

This question tests your ability to find critical values, calculate margin of error, and build a confidence interval for a mean.

Key Terms and Formulas:

• Sample mean: $\bar{x}$

• Sample standard deviation: $s$

• Sample size: $n$

• Degrees of freedom: $df = n - 1$

• Critical value: $t_{\alpha/2}$

• Margin of error: $E = t_{\alpha/2} \frac{s}{\sqrt{n}}$

• Confidence interval: $\bar{x} \pm E$

Step-by-Step Guidance

1. Find degrees of freedom: $df = 16 - 1 = 15$

2. Look up $t_{0.025}$ for $df = 15$ (for 95% confidence).

3. Calculate margin of error: $E = t_{0.025} \frac{40}{\sqrt{16}}$

4. Set up the confidence interval: $1150 \pm E$

Try solving on your own before revealing the answer!

Final Answer:

$t_{0.025,15} \approx 2.131$

Margin of error: $E = 2.131 \times \frac{40}{4} = 21.31$

Confidence interval: $1150 \pm 21.31$

Q4. Hypothesis Test for a Proportion

Background

Topic: Hypothesis testing for a population proportion using the z-test.

This question tests your ability to set up hypotheses, calculate test statistics, and interpret p-values.

Key Terms and Formulas:

• Null hypothesis: $H_0: p = 0.80$

• Alternative hypothesis: $H_1: p > 0.80$

• Test statistic: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 q_0}{n}}}$

• Significance level: $\alpha = 0.01$

Step-by-Step Guidance

1. State $H_0$ and $H_1$ in symbolic form.

2. Calculate $\hat{p} = \frac{390}{500}$ and $\hat{q} = 1 - \hat{p}$.

3. Plug values into the z-test formula: $z = \frac{\hat{p} - 0.80}{\sqrt{\frac{0.80 \times 0.20}{500}}}$

4. Find the p-value for the calculated z and compare to $\alpha$.

Try solving on your own before revealing the answer!

Final Answer:

$H_0: p = 0.80$, $H_1: p > 0.80$

$\hat{p} = 0.78$, $z \approx -0.707$

p-value $\approx 0.24$; fail to reject $H_0$ at $\alpha = 0.01$

Q5. Hypothesis Test for a Mean (t-test)

Background

Topic: Hypothesis testing for a population mean using the t-test.

This question tests your ability to check requirements, compute t-statistics, and interpret results.

Key Terms and Formulas:

• Null hypothesis: $H_0: \mu = 10$

• Alternative hypothesis: $H_1: \mu > 10$

• Test statistic: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

• Significance level: $\alpha = 0.05$

Step-by-Step Guidance

1. Check if sample is random, population is normal or $n \geq 30$, and $\sigma$ unknown.

2. Calculate $t = \frac{12.5 - 10}{3.2 / \sqrt{12}}$

3. Compare calculated t to critical value for $df = 11$ at $\alpha = 0.05$.

Try solving on your own before revealing the answer!

Final Answer:

$t \approx 2.70$

Critical value $t_{0.05,11} \approx 1.796$; since $t > t_{crit}$, reject $H_0$

The sample supports the gym's claim.

Q6. Comparing Two Independent Means

Background

Topic: Comparing means from two independent samples using the t-test for unequal variances.

This question tests your ability to calculate the test statistic and interpret p-values for two-sample comparisons.

Key Terms and Formulas:

• Test statistic: $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• Degrees of freedom: $df = 29$ (given)

Step-by-Step Guidance

1. Calculate $\bar{x}_1 - \bar{x}_2 = 14.5 - 13.8$

2. Compute the denominator: $\sqrt{\frac{1.2^2}{30} + \frac{1.5^2}{30}}$

3. Plug values into the t formula and solve for t.

4. Use $df = 29$ to find the p-value for a two-tailed test.

Try solving on your own before revealing the answer!

Final Answer:

$t \approx 2.13$

p-value $\approx 0.042$ for $df = 29$

There is a significant difference at $\alpha = 0.05$.

Q7. Comparing Two Proportions

Background

Topic: Comparing proportions from two independent samples and constructing confidence intervals for their difference.

This question tests your ability to set up hypotheses, calculate confidence intervals, and interpret significance.

Key Terms and Formulas:

• Null hypothesis: $H_0: p_1 = p_2$

• Difference in proportions: $\hat{p}_1 - \hat{p}_2$

• Confidence interval: $(\hat{p}_1 - \hat{p}_2) \pm z_{\alpha/2} \sqrt{\frac{\hat{p}_1 \hat{q}_1}{n_1} + \frac{\hat{p}_2 \hat{q}_2}{n_2}}$

Step-by-Step Guidance

1. State $H_0: p_1 = p_2$

2. Calculate $\hat{p}_1 = \frac{120}{200}$, $\hat{p}_2 = \frac{140}{250}$

3. Find $\hat{q}_1 = 1 - \hat{p}_1$, $\hat{q}_2 = 1 - \hat{p}_2$

4. Set up the confidence interval formula for the difference.

Try solving on your own before revealing the answer!

Final Answer:

$\hat{p}_1 = 0.60$, $\hat{p}_2 = 0.56$

Confidence interval: $0.04 \pm 1.96 \sqrt{\frac{0.60 \times 0.40}{200} + \frac{0.56 \times 0.44}{250}}$

The interval includes 0, so there is no significant difference.