Final Exam Review Guidance for MAT 124 (Statistics) Chapters 6–10

Q1. Probability Distribution for Busy Telephone Lines

Background

Topic: Discrete Probability Distributions

This question tests your understanding of probability distributions for discrete random variables, as well as calculation of mean and standard deviation for such distributions.

Key Terms and Formulas:

• Random Variable ($X$): Represents the number of busy telephone lines.

• Probability Distribution: $P(X=x)$ gives the probability for each value of $X$.

• Mean (Expected Value): $\mu = \sum x \cdot P(X=x)$

• Standard Deviation: $\sigma = \sqrt{\sum (x - \mu)^2 \cdot P(X=x)}$

Step-by-Step Guidance

1. For part (a), identify which values of $X$ are less than 2 (i.e., $X = 0$ and $X = 1$).

2. Add the probabilities for $X = 0$ and $X = 1$ to find $P(X < 2)$.

3. For part (b), identify the probabilities for $X = 1$ and $X = 2$ and sum them for $P(X = 1 \text{ or } X = 2)$.

4. For part (c), use the formula for the mean: $\mu = \sum x \cdot P(X=x)$, multiplying each value of $x$ by its probability and summing.

5. For part (d), use the formula for standard deviation: $\sigma = \sqrt{\sum (x - \mu)^2 \cdot P(X=x)}$, but first calculate $\mu$ from part (c).

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Q2. Binomial Probability for Weather Forecast Accuracy

Background

Topic: Binomial Distribution

This question tests your ability to apply the binomial probability formula to calculate probabilities, mean, and standard deviation for a binomial random variable.

Key Terms and Formulas:

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

• Mean: $\mu = n p$

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

• Where $n = 5$, $p = 0.8$

Step-by-Step Guidance

1. For part (a), use the binomial formula to find $P(X = 3)$, where $X$ is the number of accurate days.

2. For part (b), calculate $P(X \geq 4)$ by finding $P(X = 4)$ and $P(X = 5)$ and adding them.

3. For part (c), use the mean formula: $\mu = n p$.

4. For part (d), use the standard deviation formula: $\sigma = \sqrt{n p (1-p)}$.

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Q3. Normal Distribution for Telephone Call Lengths

Background

Topic: Normal Distribution and Probability

This question tests your ability to use the properties of the normal distribution to find probabilities and percentages for given intervals.

Key Terms and Formulas:

• Standard Normal Variable: $Z = \frac{X - \mu}{\sigma}$

• Mean ($\mu$): 10 minutes

• Standard Deviation ($\sigma$): 3 minutes

Step-by-Step Guidance

1. For part (a), calculate the $Z$-score for 16 minutes: $Z = \frac{16 - 10}{3}$.

2. Use the $Z$-score to find the probability that $X > 16$ using the standard normal table.

3. For part (b), calculate $Z$-scores for 15 and 18 minutes: $Z_{15} = \frac{15 - 10}{3}$, $Z_{18} = \frac{18 - 10}{3}$.

4. Find the area between these two $Z$-scores using the standard normal table.

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Q4. Assessing Normality of Basketball Player Heights

Background

Topic: Assessing Normality (Normal Probability Plot, Boxplot, Correlation Coefficient)

This question tests your ability to use graphical and statistical methods to assess whether a sample comes from a normal distribution.

Key Terms and Formulas:

• Normal Probability Plot: Plots sample data against theoretical normal quantiles.

• Boxplot: Visualizes spread, center, and outliers.

• Correlation Coefficient: Measures linearity between normal scores and sample values.

Step-by-Step Guidance

1. Arrange the sample data in ascending order: 66, 70, 72, 75, 77.

2. Construct a normal probability plot by pairing each data value with its corresponding normal quantile.

3. Draw a boxplot to visualize the distribution and check for symmetry and outliers.

4. Calculate the correlation coefficient between the normal scores and the sample values.

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Q5. SAT Scores and the Empirical Rule

Background

Topic: Empirical Rule for Normal Distributions

This question tests your understanding of the empirical rule (68-95-99.7 rule) for normal distributions.

Key Terms and Formulas:

• Empirical Rule: For a normal distribution, about 68% of values fall within 1 standard deviation, 95% within 2, and 99.7% within 3.

• Mean ($\mu$): 1643

• Standard Deviation ($\sigma$): 341.3

Step-by-Step Guidance

1. Calculate the range for 68%: $\mu \pm 1\sigma$.

2. Calculate the range for 95%: $\mu \pm 2\sigma$.

3. Calculate the range for 99.7%: $\mu \pm 3\sigma$.

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Q6. Sampling Distribution of Teacher Salaries

Background

Topic: Central Limit Theorem and Sampling Distribution

This question tests your ability to use the sampling distribution of the sample mean to find probabilities.

Key Terms and Formulas:

• Sampling Distribution Mean: $\mu_{\bar{x}} = \mu$

• Sampling Distribution Standard Deviation: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Probability: Use $Z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}}$

Step-by-Step Guidance

1. Calculate the standard error: $\sigma_{\bar{x}} = \frac{7200}{\sqrt{64}}$.

2. Find the $Z$-scores for $\bar{x} = 56,300 \pm 1,000$.

3. Use the standard normal table to find the probability that the sample mean falls within this range.

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Q7. Confidence Interval for Calculator Life Expectancy

Background

Topic: Confidence Intervals for Means

This question tests your ability to construct and interpret a confidence interval for a population mean using sample data.

Key Terms and Formulas:

• Confidence Interval: $\bar{x} \pm z^* \frac{s}{\sqrt{n}}$

• Sample Mean ($\bar{x}$): 21 months

• Sample Standard Deviation ($s$): 2.8 months

• Sample Size ($n$): 100

• $z^*$ for 95% confidence: approximately 1.96

Step-by-Step Guidance

1. Calculate the standard error: $SE = \frac{2.8}{\sqrt{100}}$.

2. Multiply the standard error by $z^*$ to find the margin of error.

3. Add and subtract the margin of error from the sample mean to get the confidence interval.

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Q8. Hypothesis Test for Iron Intake

Background

Topic: Hypothesis Testing for Means (One-Sample t-Test)

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

Key Terms and Formulas:

• Null Hypothesis ($H_0$): $\mu = 18$ mg

• Alternative Hypothesis ($H_a$): $\mu < 18$ mg

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

• Sample Mean ($\bar{x}$): 14.10 mg

• Sample Standard Deviation ($s$): 4.2 mg

• Sample Size ($n$): 10

Step-by-Step Guidance

1. State the null and alternative hypotheses.

2. Calculate the test statistic using the formula above.

3. Determine the critical value for a one-tailed test at the 1% significance level with $n-1$ degrees of freedom.

4. Compare the test statistic to the critical value to decide whether to reject $H_0$.

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Q9. Type of Error for Iron Intake Hypothesis Test (If True Mean is 18 mg)

Background

Topic: Type I and Type II Errors

This question tests your understanding of errors in hypothesis testing.

Key Terms and Formulas:

• Type I Error: Rejecting $H_0$ when it is true.

• Type II Error: Failing to reject $H_0$ when it is false.

Step-by-Step Guidance

1. Recall the decision made in Q8 (reject or fail to reject $H_0$).

2. Given that the true mean is 18 mg, determine whether the decision was correct, a Type I error, or a Type II error.

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Q10. Type of Error for Iron Intake Hypothesis Test (If True Mean is Less Than 18 mg)

Background

Topic: Type I and Type II Errors

This question tests your understanding of errors in hypothesis testing.

Key Terms and Formulas:

• Type I Error: Rejecting $H_0$ when it is true.

• Type II Error: Failing to reject $H_0$ when it is false.

Step-by-Step Guidance

1. Recall the decision made in Q8 (reject or fail to reject $H_0$).

2. Given that the true mean is less than 18 mg, determine whether the decision was correct, a Type I error, or a Type II error.

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Q11. Hypothesis Test for Mean Height of Women

Background

Topic: Hypothesis Testing for Means (One-Sample t-Test)

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

Key Terms and Formulas:

• Null Hypothesis ($H_0$): $\mu = 62.3$ in

• Alternative Hypothesis ($H_a$): $\mu < 62.3$ in

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

• Sample Mean ($\bar{x}$): 61.1 in

• Sample Standard Deviation ($s$): 3.6 in

• Sample Size ($n$): 100

Step-by-Step Guidance

1. State the null and alternative hypotheses.

2. Calculate the test statistic using the formula above.

3. Determine the critical value for a one-tailed test at the 5% significance level with $n-1$ degrees of freedom.

4. Compare the test statistic to the critical value to decide whether to reject $H_0$.

Try solving on your own before revealing the answer!

Q12. Type of Error for Height Hypothesis Test (If True Mean is 62.3 in)

Background

Topic: Type I and Type II Errors

This question tests your understanding of errors in hypothesis testing.

Key Terms and Formulas:

• Type I Error: Rejecting $H_0$ when it is true.

• Type II Error: Failing to reject $H_0$ when it is false.

Step-by-Step Guidance

1. Recall the decision made in Q11 (reject or fail to reject $H_0$).

2. Given that the true mean is 62.3 in, determine whether the decision was correct, a Type I error, or a Type II error.

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Q13. Type of Error for Height Hypothesis Test (If True Mean is 61 in)

Background

Topic: Type I and Type II Errors

This question tests your understanding of errors in hypothesis testing.

Key Terms and Formulas:

• Type I Error: Rejecting $H_0$ when it is true.

• Type II Error: Failing to reject $H_0$ when it is false.

Step-by-Step Guidance

1. Recall the decision made in Q11 (reject or fail to reject $H_0$).

2. Given that the true mean is 61 in, determine whether the decision was correct, a Type I error, or a Type II error.

Try solving on your own before revealing the answer!