Probability and Statistics Final Review Guidance

Q1. Using a random sample of 15 households, find:

• Probability that 10 households have a DVD player: $P(X = 10)$

• Probability that at least 1 household has a DVD player: $P(X \geq 1)$

• Mean of this binomial distribution

• Standard deviation of this binomial distribution

Background

Topic: Binomial Distribution

This question tests your understanding of binomial probability, including calculating specific probabilities, mean, and standard deviation for a binomial random variable.

Key Terms and Formulas

• Binomial Probability Formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• Mean: $\mu = np$

• Standard Deviation: $\sigma = \sqrt{np(1-p)}$

• Where:

  • $n$ = number of trials (households sampled)

  • $p$ = probability of success (household has DVD player)

  • $k$ = number of successes

Step-by-Step Guidance

1. Identify the parameters: $n = 15$, $p = 0.70$.

2. For $P(X = 10)$, use the binomial formula: $P(X = 10) = \binom{15}{10} (0.70)^{10} (0.30)^{5}$.

3. For $P(X \geq 1)$, recall that $P(X \geq 1) = 1 - P(X = 0)$. Calculate $P(X = 0)$ using the binomial formula.

4. To find the mean, use $\mu = np$.

5. To find the standard deviation, use $\sigma = \sqrt{np(1-p)}$.

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Q2. Mean hypothesis testing: $H_0: \mu = 12$, $H_1: \mu < 12$, p-value = 0.003, $\alpha = 0.01$. Which statement is true?

Background

Topic: Hypothesis Testing

This question tests your ability to interpret hypothesis test results, specifically the meaning of p-values and the direction of the test.

Key Terms

• Null Hypothesis ($H_0$): The population mean equals 12.

• Alternative Hypothesis ($H_1$): The population mean is less than 12.

• p-value: Probability of observing the sample result, or more extreme, if $H_0$ is true.

• Significance Level ($\alpha$): Threshold for rejecting $H_0$.

Step-by-Step Guidance

1. Recognize that $H_1$ is a one-tailed (left-tailed) test.

2. Compare the p-value (0.003) to $\alpha$ (0.01).

3. Recall: If p-value $<$ $\alpha$, reject $H_0$.

4. Consider what rejecting $H_0$ means in context of the hypotheses.

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Q3. Which definition states that as sample size increases, the distribution of sample means approaches a normal distribution?

Background

Topic: Sampling Distributions

This question tests your knowledge of foundational statistical theorems about sampling distributions.

Key Terms

• Central Limit Theorem (CLT): As $n$ increases, the sampling distribution of the sample mean becomes approximately normal.

• Law of Large Numbers: As $n$ increases, sample mean approaches population mean.

Step-by-Step Guidance

1. Review the definitions of each option.

2. Identify which theorem specifically addresses the normality of sample means as $n$ increases.

3. Recall that the CLT applies regardless of the population's original distribution.

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Q4. When calculating sample size to achieve a certain error, which value of $\hat{p}$ leads to a larger sample?

Background

Topic: Sample Size Calculation for Proportions

This question tests your understanding of how the estimated proportion affects required sample size.

Key Formula

• Sample size formula: $n = \frac{z^2 \hat{p}(1-\hat{p})}{E^2}$

• Where:

  • $z$ = z-score for desired confidence level

  • $\hat{p}$ = estimated proportion

  • $E$ = margin of error

Step-by-Step Guidance

1. Recall that $\hat{p}(1-\hat{p})$ is maximized when $\hat{p} = 0.5$.

2. Compare the values given: $\hat{p} = 0.15, 0.45, 0.85, 1$.

3. Calculate $\hat{p}(1-\hat{p})$ for each option to see which is largest.

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Q5. If we want to increase the precision of a confidence interval, what should we do?

Background

Topic: Confidence Intervals

This question tests your understanding of factors affecting the width (precision) of a confidence interval.

Key Terms

• Precision: Narrower confidence interval means higher precision.

• Margin of Error: $E = z \frac{\sigma}{\sqrt{n}}$ or $E = z \frac{s}{\sqrt{n}}$

Step-by-Step Guidance

1. Recall that increasing sample size ($n$) decreases margin of error.

2. Consider how changes in mean, standard deviation, and margin of error affect interval width.

3. Identify which option directly increases precision.

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Q6. Estimate the true proportion of Americans who celebrate Valentine’s Day using a 98% confidence interval, given 2200 out of 4000 sampled.

Background

Topic: Confidence Interval for Proportions

This question tests your ability to construct a confidence interval for a population proportion.

Key Formula

• Sample proportion: $\hat{p} = \frac{x}{n}$

• Confidence interval: $\hat{p} \pm z \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• $z$ = z-score for 98% confidence

Step-by-Step Guidance

1. Calculate $\hat{p}$ using $x = 2200$, $n = 4000$.

2. Find the z-score for 98% confidence (look up or calculate).

3. Compute the standard error: $\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.

4. Set up the confidence interval formula.

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Q7. SAT Reading scores: mean = 510, SD = 70, normal distribution. Find:

• Probability student scored below 490

• Probability student scored above 590

• Probability student scored between 490 and 590

Background

Topic: Normal Distribution and Probability

This question tests your ability to use the normal distribution to find probabilities for specific score ranges.

Key Formula

• Z-score: $z = \frac{x - \mu}{\sigma}$

• Use standard normal tables or calculator for probabilities.

Step-by-Step Guidance

1. Calculate z-scores for 490 and 590 using $\mu = 510$, $\sigma = 70$.

2. For each probability, use the z-score to find the corresponding area under the normal curve.

3. For "between" probability, subtract the lower area from the higher area.

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Q8. A 99% confidence interval about the mean is constructed with a length of 9; what is the margin of error?

Background

Topic: Margin of Error in Confidence Intervals

This question tests your understanding of how margin of error relates to the length of a confidence interval.

Key Formula

• Length of confidence interval = $2 \times$ margin of error

• Margin of error: $E = z \frac{\sigma}{\sqrt{n}}$ or $E = z \frac{s}{\sqrt{n}}$

Step-by-Step Guidance

1. Recall that the confidence interval is $\bar{x} \pm E$, so total length is $2E$.

2. Set up the equation: $2E = 9$.

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Q9. World Cup finals goals data:

• Compute and interpret the mean of the random variable $x$

• Compute the standard deviation and variance of $x$

• Probability that teams score 6 goals

• Probability that teams score two or more goals

Background

Topic: Discrete Probability Distributions

This question tests your ability to compute mean, variance, and probabilities for a discrete random variable using frequency data.

Key Formulas

• Mean: $\mu = \sum x P(X = x)$

• Variance: $\sigma^2 = \sum x^2 P(X = x) - \mu^2$

• Standard deviation: $\sigma = \sqrt{\sigma^2}$

Step-by-Step Guidance

1. Calculate $P(X = x)$ for each $x$ using frequencies.

2. Compute $\mu$ by summing $x P(X = x)$.

3. Compute $\sum x^2 P(X = x)$ for variance.

4. Set up variance and standard deviation formulas.

5. For probabilities, sum $P(X = x)$ for relevant $x$ values.

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Q10. Test the claim about population proportion $p = 0.250$ given $n = 48$ and $\hat{p} = 0.231$, $\alpha = 0.01$.

Background

Topic: Hypothesis Test for Proportion

This question tests your ability to perform a hypothesis test for a population proportion.

Key Formula

• Test statistic: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

• Where $p_0$ is the claimed proportion.

Step-by-Step Guidance

1. State null and alternative hypotheses: $H_0: p = 0.250$, $H_1: p \neq 0.250$ (or specify direction).

2. Calculate the test statistic using the formula above.

3. Find the p-value corresponding to the calculated $z$.

4. Compare p-value to $\alpha$ to decide whether to reject $H_0$.

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Q11. Test the claim that mean waiting time is less than 5 minutes ($n = 20$, $\bar{x} = 3.5$, $s = 2.1$, $\alpha = 0.01$).

• State Null and Alternative Hypotheses

• State level of alpha and calculate test statistic

• Find the P-value

• Reject $H_0$ or Do Not Reject $H_0$

• State your conclusion

Background

Topic: One-Sample t-Test

This question tests your ability to perform a hypothesis test for a population mean using sample data.

Key Formula

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

• Degrees of freedom: $df = n - 1$

Step-by-Step Guidance

1. State $H_0: \mu = 5$, $H_1: \mu < 5$.

2. Calculate the test statistic using the formula above.

3. Find the p-value using the t-distribution with $df = 19$.

4. Compare p-value to $\alpha$ to decide whether to reject $H_0$.

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Q12. To minimize sample size for proportions, if an old study used a proportion between 0.1 and 0.15, which value should the researcher use?

Background

Topic: Sample Size Calculation for Proportions

This question tests your understanding of how the estimated proportion affects required sample size.

Key Formula

• Sample size formula: $n = \frac{z^2 p(1-p)}{E^2}$

Step-by-Step Guidance

1. Recall that $p(1-p)$ is minimized when $p$ is closest to 0 or 1.

2. Compare $p = 0.1$ and $p = 0.15$ to see which yields a smaller $p(1-p)$.

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Q13. What conditions from a box-plot and normality plot indicate data is normally distributed?

Background

Topic: Assessing Normality

This question tests your ability to interpret graphical displays to assess normality of data.

Key Terms

• Box-plot: Symmetry, absence of outliers, equal whisker lengths suggest normality.

• Normality plot (Q-Q plot): Points lie close to a straight line.

Step-by-Step Guidance

1. Examine box-plot for symmetry and outliers.

2. Check normality plot for linearity.

3. Combine evidence from both plots to assess normality.

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Q14. Airline claims no-show rate is less than 5%. In a sample of 420 reservations, 19 were no-shows. At $\alpha = 0.01$, test the claim. Round $\hat{p}$ to nearest thousandth for test statistic.

Background

Topic: Hypothesis Test for Proportion

This question tests your ability to perform a hypothesis test for a population proportion.

Key Formula

• Sample proportion: $\hat{p} = \frac{19}{420}$

• Test statistic: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

Step-by-Step Guidance

1. State $H_0: p = 0.05$, $H_1: p < 0.05$.

2. Calculate $\hat{p}$ and round to nearest thousandth.

3. Compute the test statistic using the formula above.

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

Try solving on your own before revealing the answer!