Q1. From a group of 5 students (A, B, C, D, E), three will be chosen.

Background

Topic: Counting, Probability, and Sample Spaces

This question tests your understanding of sample spaces, combinations, and basic probability concepts.

Key Terms and Formulas:

• Sample Space: The set of all possible outcomes.

• Combination Formula:

• Probability:

Step-by-Step Guidance

1. List all possible groups of 3 students chosen from 5. Use the combination formula to determine how many groups there are:

2. For part (b), count how many groups include D. Consider D as one of the chosen, then select 2 more from the remaining 4 students.

3. For part (c), count how many groups do not include D. Select 3 students from the remaining 4.

4. For part (d), count how many groups include both A and E. With A and E chosen, select 1 more from the remaining 3 students.

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Q2. Given a frequency distribution of siblings, find the probability of having fewer than two siblings.

Background

Topic: Probability from Frequency Distributions

This question tests your ability to calculate probabilities using frequency tables.

Key Terms and Formulas:

• Probability:

Step-by-Step Guidance

1. Identify the frequencies for 0 and 1 siblings: 35 and 95.

2. Add these frequencies to get the total number of students with fewer than two siblings.

3. Sum all frequencies to get the total number of students surveyed.

4. Set up the probability formula:

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Q3. A survey of 100 MSU students: 65 support the governor’s policies (43 male), 35 do not (21 male). Find:

Background

Topic: Contingency Tables and Probability

This question tests your ability to organize data in a contingency table and calculate probabilities.

Key Terms and Formulas:

• Contingency Table: Organizes data by categories.

• Probability:

• Union:

Step-by-Step Guidance

1. Fill in the contingency table: Support/Do not support vs. Male/Female.

2. Calculate the number of females: subtract males from totals in each category.

3. For (a), sum all females and divide by 100.

4. For (b), use the union formula: count males and those who do not support, subtract overlap.

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Q4. Given a probability distribution for a discrete random variable:

Background

Topic: Probability Distributions, Expected Value, and Standard Deviation

This question tests your ability to work with discrete probability distributions and calculate expected value and standard deviation.

Key Terms and Formulas:

• Expected Value:

• Standard Deviation:

Step-by-Step Guidance

1. For (a), sum probabilities for (i.e., and ).

2. For (b), multiply each value by its probability and sum to find .

3. For (c), use the expected value from (b) and calculate for each , multiply by probability, sum, and take the square root.

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Q5. In a survey, 69% use Yahoo as their primary search engine. Sample of 10 students:

Background

Topic: Binomial Probability Distribution

This question tests your ability to apply the binomial distribution to calculate probabilities, expected value, and standard deviation.

Key Terms and Formulas:

• Binomial Probability:

• Expected Value:

• Standard Deviation:

Step-by-Step Guidance

1. For (a), use , , in the binomial formula.

2. For (b), sum probabilities for and using the binomial formula.

3. For (c), calculate expected value: .

4. For (d), calculate standard deviation: .

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Q6. Use the standard normal table to determine:

Background

Topic: Standard Normal Distribution and Z-Scores

This question tests your ability to use the standard normal table to find probabilities and z-values.

Key Terms and Formulas:

• Z-score:

• Standard Normal Table: Used to find or .

Step-by-Step Guidance

1. For (a), look up in the table.

2. For (b), find and , then subtract to get the probability between.

3. For (c), find by subtracting from 1.

4. For (d), find the z-value where (inverse lookup).

5. For (e), find (the z-value with 2% in the right tail).

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Q7. Scores on a standardized test are normally distributed with mean 24 and standard deviation 3.5.

Background

Topic: Normal Distribution, Empirical Rule, and Percentiles

This question tests your ability to apply the normal distribution, calculate probabilities, and use the Empirical Rule.

Key Terms and Formulas:

• Z-score:

• Empirical Rule: 68%, 95%, 99.74% within 1, 2, 3 standard deviations.

• Percentile: Use the standard normal table to find the score corresponding to a given percentile.

Step-by-Step Guidance

1. For (a), calculate the z-score for 25, then use the normal table to find the percent scoring 25 or higher.

2. For (b), calculate z-scores for 21 and 26, then find the probability between them.

3. For (c), use the Empirical Rule to find the range for 99.74%: .

4. For (d), find the z-score for the 83rd percentile, then convert to the test score using .

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