# The Distribution of Scores

1. 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?

2. Over the past 10 years the local school district has measured physical fitness for all high school freshmen. During that time, the average score on a treadmill endurance task has been µ=19.8 minutes with a standard deviation of σ=7.2 minutes. Assuming that the distribution is approximately normal, find each of the following probabilities.

a. What is the probability of randomly selecting a student with a treadmill time greater than 25 minutes? In symbols, p(X>25)= ?

b. What is the probability of randomly selecting a student with a time greater than 30 minutes? In symbols, p(x>30) ?

c. If the school required a minimum time of 10 minutes for students to pass the physical education course, what proportion of the freshman would fail?

3. Rochester, New York, averages µ=21.9 inches of snow for the month of December. The distribution of snowfall amounts is approximately normal with a standard deviation of σ=6.5 inches. This year a local jewelry store is advertising a refund of 50% off all purchases made in December, if we finish the month with more than 3 feet (36 inches) of total snowfall. What is the probability that the jewelry store will have to pay off on its promise?

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1. 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?

Solution:

a)

Therefore, a SAT score of 604 separates the top 15% of the distribution from the rest.

b)

Therefore, a SAT score of 584 separates the ...

#### Solution Summary

The distribution of scores for SAT x-values are given.

Simple Statistics Problems (Entry Level)

4. For a population with μ=50 and σ=10.

a. Find the z-score for each of the following X values. (Note: You should be able to find these values using the definition of a z-score. You should not need to use a formula or do any serious calculations.)

X = 55 X = 60 X = 75

X = 45 X = 30 X = 35

b. Find the score (X value) that corresponds to each of the following z-scores. (Note: You should be able to find these values using the definition of a z-score. (Again, you should be able to find these values without any formula or serious calculations.)

z = 1.00 z = 0.80 z = 1.50

z = -0.50 z = -0.30 z = -1.50

10. Find the z-score corresponding to a score of X = 60 for each of the following distributions.

a. μ = 50 and σ = 10

b. μ = 50 and σ = 5

c. μ = 70 and σ = 20

d. μ = 70 and σ = 5

12. A score that is 6 points below the mean corresponds to a z-score of z = -2.00. What is the population standard deviation?

20. For each of the following populations, would a score of X = 50 be considered a central score (near the middle of the distribution) or an extreme score (far out in the tail of the distribution)?

a. μ = 45 and σ = 10

b. μ = 40 and σ = 2

c. μ = 55 and σ = 2

d. μ = 60 and σ = 20

26. A distribution with a mean of μ = 38 and a standard deviation of σ = 20 is being transformed into a standardized distribution with μ = 50 and σ = 10. Find the new, standardized score for each of the following values from the original population.

a. X = 48 b. X = 40

c. X = 30 d. X = 18

28. A population consists of the following N = 5 scores: 0, 6, 4, 3, and 12.

a. Compute μ and σ for the population.

b. Find the z-score for each score in the population.

c. Transform the original population into a new population of N = 5 scores with a mean of μ = 60 and a standard deviation of σ =8.

2. A kindergarten class consists of 14 boys and 11 girls. If the teacher selects children from the class using random sampling,

a. What is the probability that the first child selected will be a girl?

b. If the teacher selects a random sample of n = 3 boys, and the first two children are both boys, what is the probability that the third child selected will be a girl?

8. Find each of the following probabilities for a normal distribution.

a. p(z > -1.00)

b. p(z > -0.80)

c. p(z < 0.25)

d. p(z < 1.25)

12. 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

16. The distribution of IQ 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?

22. 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 simply by guessing?

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

26. A trick coin has been weighted so that heads occurs with the 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?

Chapter 7

2. Describe the distribution of sample means (shape, expected value, and standard error) for samples of n = 36 selected from a population with a mean of µ = 100 and a standard deviation of σ = 12.

8. If the population standard deviation is σ = 20, how large a sample is necessary to have a standard error that is

a. Less than 4 points?

b. Less than 2 points?

c. Less than 1 point?

10. For a population with a mean of μ = 60 and a standard deviation of σ = 24, find the z-score corresponding to each of the following samples.

a. M = 63 for a sample of n = 16 scores.

b. M = 63 for a sample of n = 36 scores.

c. M = 63 for a sample of n = 64 scores.

16. A population of scores forms a normal distribution with a mean of μ = 40 and a standard deviation of σ =12.

a. What is the probability of randomly selecting a score less than X = 34?

b. What is the probability of selecting a sample of n = 9 scores with a mean less than M = 34?

c. What is the probability of selecting a sample of n = 36 scores with a mean less than M = 34?

20. A normal-shaped distribution has a μ = 80 and a standard deviation of σ =15.

a. Sketch the distribution of sample means for samples of n = 25 from this population

b. What are the z-score values that form the boundaries for the middle 95% of the distribution of sample means?

c. Compute the z-score for M = 89 for a sample of n = 25 scores. Is this sample mean in the middle 95% of the distribution?

22. People are selected to serve on juries by randomly picking names from the list of registered voters. The average age for registered voters in the county is μ = 39.7 years with a standard deviation of σ = 12.4. A statistician randomly selects a sample of n = 16 people who are currently serving on juries. The average age for the individuals in the sample is M = 48.9 years.

a. How likely is it to obtain a random sample of n = 16 jurors with an average age greater than or equal to 48.9?

b. Is it reasonable to conclude that this set of n = 16 people is not a representative sample of registered voters?