# Normal Probability & Z score

1. For a normal distribution, find the z-score values that separate

A. The middle 60% of the distribution from the 40% in the tails.

B. The middle 70% of the distribution from the 30% in the tails.

C. The middle 80% of the distribution from the 20% in the tails.

D. The middle 90% of the distribution from the 10% in the tails.

2. The distribution of SAT scores is normal with µ=500 and σ = 100.

A. What SAT score, X value, separates the top 15% of the distribution from the rest?

B. What SAT score, X value, separates the top 20% of the distribution from the rest?

C. What SAT score, X value, separates the top 25% of the distribution from the rest?

3. A normal distribution has a mean of µ = 80 and a standard deviation of σ = 10. For each of the following scores, indicate whether the body of the distribution is to the left or right of the score, and find the proportion of the distribution in the body.

A. X = 85

B. X = 92

C. X = 78

D. X = 72

4. A consumer survey indicates that the average household spends µ = $155 on groceries each week. The distribution of spending amounts is approximately normal with a standard deviation of σ = $25. Based on this distribution,

A. What proportion of the population spends more than $175 per week on groceries?

B. What is the probability of randomly selecting a family that spends less than $100 per week on groceries?

C. How much money do you need to spend on groceries each week to be in the top 20% of the distribution?

5. A multiple-choice test has 48 questions, each with four response choices. If a student is simply guessing at the answers,

A. What is the probability of guessing correctly for any question?

B. On average, how many questions would a student get correct for the entire test?

C. What is the probability that a student would get more than 18 answers correct by guessing?

D. What is the probability that a student would get 18 or more answers correct by guessing?

6. A trick coin has been weighted so that heads occurs with a probability of p = 2/3, and p(tails) = 1/3. If you toss this coin 72 times:

A. How many heads would you expect to get on average?

B. What is the probability of getting more than 50 heads?

C. What is the probability of getting exactly 50 heads?

PLEASE SHOW STEP BY STEP HOW YOU SOLVE THESE PROBLEMS.

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#### Solution Summary

The solution provides step by step method for the calculation of probability using the Z score. Formula for the calculation and Interpretations of the results are also included.

Normal probability z-scores

If a student had a z-score of 1, what would be the raw score (rating)?

If a student had a z-score of 1, what percentage of the participants would be

expected to rate the product higher than he/she did?

If a student gave a rating of 75, what would be the z-score?

What is the probability that a rating is below 51.03?

What is the probability that a rating is higher than 80?

What percentage of participants would be expected to rate the product lower than 75?

What is the probability that a rating falls between 25 and 65?

What is the probability that a rating falls between 75 and 80?

If the z-score is 2, what is the rating?

If the rating is 21, what is the z-score?

What is the probability of observing a rating lower than 20 or higher than 80?

4. What if one participant gave a particularly odd (extreme) rating? How extreme would it have to be for us to suspect that it should be discounted (i.e., the participant was rating a different product or was trying to ruin our data or wasn't really paying attention to the task)?

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