Step-by-Step Guidance for Statistics Exam Review Problems

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Q1. Probability Distribution for Number of Pizzas Delivered

Background

Topic: Discrete Probability Distributions

This question tests your understanding of probability distributions, expected value, and standard deviation for discrete random variables.

Key Terms and Formulas:

Probability Distribution: A table showing the probabilities for each possible value of a random variable.
Expected Value:
Standard Deviation:

Step-by-Step Guidance

For part (a), recall that the sum of all probabilities in a distribution must equal 1. Set up the equation using the given probabilities and solve for .
For part (b), identify which values of satisfy and sum their probabilities.
For part (c), use the expected value formula: multiply each value of by its probability and sum the results.
For part (d), use the standard deviation formula: calculate for each , multiply by , sum, and take the square root.

Try solving on your own before revealing the answer!

Final Answers:

Each part uses the appropriate formula for discrete probability distributions.

Q2. Probability for Teacher Salaries (Normal Distribution)

Background

Topic: Normal Distribution and Probability

This question tests your ability to use the normal distribution to find probabilities for values above or below a certain threshold.

Key Terms and Formulas:

Normal Distribution: A continuous probability distribution characterized by mean and standard deviation .
Z-score:
Probability: Use the standard normal table to find or .

Step-by-Step Guidance

For part (a), calculate the z-score for using the formula .
Use the z-score to find the probability that by looking up the value in the standard normal table.
For part (b), calculate the z-score for using .
Use the z-score to find the probability that .

Try solving on your own before revealing the answer!

Final Answers:

These probabilities are found using the z-score and the standard normal table.

Q3. Police Academy Cutoff Score (Top 20%)

Background

Topic: Normal Distribution Percentiles

This question tests your ability to find a cutoff value corresponding to a given percentile in a normal distribution.

Key Terms and Formulas:

Percentile: The value below which a given percentage of observations fall.
Z-score for top 20%: Find such that .
Cutoff score:

Step-by-Step Guidance

Find the z-score corresponding to the top 20% (i.e., the 80th percentile).
Use the z-score formula to solve for the cutoff score: .
Look up the z-score for the 80th percentile in the standard normal table.

Try solving on your own before revealing the answer!

Final Answer:

The cutoff score is approximately 71.6.

This is found by using the z-score for the 80th percentile and plugging it into the formula.

Q4. Probability of Business Failures (Binomial Distribution)

Background

Topic: Binomial Distribution

This question tests your ability to use the binomial distribution to find probabilities and expected values.

Key Terms and Formulas:

Binomial Probability:
Mean:

Step-by-Step Guidance

For part (a), use , , in the binomial formula.
For part (b), use the complement rule: .
For part (c), calculate the mean: .

Try solving on your own before revealing the answer!

Final Answers:

a)
b)
c) Mean number of failures: $6$

These are calculated using the binomial formula and the complement rule.

Q5. Probability All Tomatoes Sold (Normal Distribution)

Background

Topic: Normal Distribution Probability

This question tests your ability to find the probability that a value is less than or equal to a given threshold in a normal distribution.

Key Terms and Formulas:

Z-score:
Probability:

Step-by-Step Guidance

Calculate the z-score for .
Use the z-score to find the probability from the standard normal table.

Try solving on your own before revealing the answer!

Final Answer:

This is a very low probability, indicating it's unlikely all tomatoes will be sold.

Q6. Probability for Precipitation (Normal Distribution and Sampling)

Background

Topic: Normal Distribution and Central Limit Theorem

This question tests your ability to find probabilities for individual values and sample means using the normal distribution.

Key Terms and Formulas:

Z-score for individual:
Z-score for sample mean:

Step-by-Step Guidance

For part (a), calculate the z-score for .
Find using the standard normal table.
For part (b), calculate z-scores for and with .
Find the probability that the sample mean is between these values.

Try solving on your own before revealing the answer!

Final Answers:

These are found using z-scores and the normal table.

Q7. Properties of Normal Distribution and Empirical Rule

Background

Topic: Properties of Normal Distribution and Empirical Rule

This question tests your knowledge of the characteristics of the normal curve and the Empirical Rule.

Key Terms:

Normal Distribution: Symmetrical, bell-shaped curve.
Empirical Rule: About 95% of data lies within 2 standard deviations of the mean.

Step-by-Step Guidance

List two unique properties: symmetry about the mean, and total area under the curve equals 1.
Recall the Empirical Rule for 2 standard deviations.
Draw two normal curves with equal means but different standard deviations (one wider, one narrower).

Try solving on your own before revealing the answer!

Final Answers:

a) Symmetrical about the mean; area under the curve is 1.
b) Approximately 95% of data lies within 2 standard deviations.
c) Graphs: One curve is wider (larger standard deviation), one is narrower (smaller standard deviation), both centered at the same mean.

Q8. Confidence Interval for Population Proportion

Background

Topic: Confidence Intervals for Proportions

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

Key Terms and Formulas:

Sample proportion:
Confidence interval:

Step-by-Step Guidance

Calculate the sample proportion: .
Find the critical z-value for 95% confidence ().
Calculate the margin of error using the formula.
Set up the confidence interval using the sample proportion and margin of error.

Try solving on your own before revealing the answer!

Final Answer:

95% confidence interval:

This interval estimates the proportion of athletes who train every day.

Q9. Confidence Interval for Population Standard Deviation

Background

Topic: Confidence Intervals for Standard Deviation (Chi-Square Distribution)

This question tests your ability to construct a confidence interval for the population standard deviation using the chi-square distribution.

Key Terms and Formulas:

Confidence interval for :
,

Step-by-Step Guidance

Find the chi-square values for at the 99% confidence level.
Plug the sample standard deviation and sample size into the formula.
Calculate the lower and upper bounds for the confidence interval.

Try solving on your own before revealing the answer!

Final Answer:

99% confidence interval:

This interval estimates the population standard deviation for bookcase heights.

Q10. Free Throw Probabilities (Binomial Distribution)

Background

Topic: Binomial Distribution

This question tests your ability to use the binomial distribution to find probabilities and expected values.

Key Terms and Formulas:

Binomial Probability:
Mean:

Step-by-Step Guidance

For part (a), use , , in the binomial formula.
For part (b), use the complement rule: .
For part (c), find by summing and .
For part (d), calculate the mean: .

Try solving on your own before revealing the answer!

Final Answers:

a)
b)
c)
d) Mean number made: $8$

These are calculated using the binomial formula and complement rule.

Q11. Raffle Probability Distribution for Net Earnings

Background

Topic: Probability Distribution for Net Earnings

This question tests your ability to construct a probability distribution for net earnings based on possible outcomes and their probabilities.

Key Terms and Formulas:

Net earnings: Prize minus ticket cost.
Probability:

Step-by-Step Guidance

Calculate net earnings for each prize: , , , and for no prize.
Find the probability for each outcome: , , , and .
Construct the probability distribution table.

Try solving on your own before revealing the answer!

Final Answer:

Net earnings and probabilities:

$997
$297
$7
with probability

This table shows the probability distribution for Gaby's net earnings.

Q12. Mean for Soft-Drink Machine (Overflow Probability)

Background

Topic: Normal Distribution and Setting Mean for Desired Probability

This question tests your ability to set the mean of a normal distribution so that a certain threshold is exceeded only a specified percentage of the time.

Key Terms and Formulas:

Overflow probability:
Z-score:
Find such that

Step-by-Step Guidance

Find the z-score corresponding to the upper 3% in the normal distribution.
Set up the equation and solve for .

Try solving on your own before revealing the answer!

Final Answer:

ounces

This mean ensures only 3% of cups overflow.

Q13. Probability of Coin Tosses (Binomial Distribution)

Background

Topic: Binomial Distribution for Compound Events

This question tests your ability to use the binomial distribution for the probability of a certain number of compound events.

Key Terms and Formulas:

Probability all coins are heads:
Binomial Probability:

Step-by-Step Guidance

Calculate the probability of all 3 coins landing heads in one toss.
Use the binomial formula with , , .
Set up the calculation for .

Try solving on your own before revealing the answer!

Final Answer:

This is the probability that exactly 3 out of 10 tosses result in all coins landing heads.

Q14. Confidence Interval and Sample Size for Thunderstorm Speeds

Background

Topic: Confidence Intervals and Sample Size Calculation

This question tests your ability to construct confidence intervals for the mean, interpret changes in interval width, and calculate minimum sample size.

Key Terms and Formulas:

Confidence interval for mean:
Sample size formula:

Step-by-Step Guidance

For part (a), find the critical t-value for at 99% confidence.
Calculate the margin of error and set up the confidence interval.
For part (b), recall that increasing sample size decreases interval width, and increasing confidence level increases interval width.
For part (c), use the sample size formula with for 90% confidence, , and .

Try solving on your own before revealing the answer!

Final Answers:

a) 99% confidence interval:
b) i) Decreases; ii) Increases
c) Minimum sample size:

These are calculated using the t-distribution and sample size formula.

Q15. Soda Bottle Fill Amounts (Normal Distribution)

Background

Topic: Normal Distribution and Percentiles

This question tests your ability to find values corresponding to percentiles and use normal distribution probabilities.

Key Terms and Formulas:

Percentile: Find such that
Z-score:
For part (b):

Step-by-Step Guidance

For part (a), find the z-score corresponding to the top 4% and solve for using .
For part (b), calculate the z-score for and find the probability using the normal table.
Set up the equation and solve for .

Try solving on your own before revealing the answer!

Final Answers:

a) Fill amount: ounces
b) bottles

These are found using z-scores and normal distribution probabilities.