Question 1

IQ test scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the x-score that corresponds to a z-score of -1.645.

Accepted Answer

The x-score is calculated as the z-score multiplied by the standard deviation and added to the mean. So, x = z * s + m = -1.645 * 15 + 100 = 75.3.

Question 2

In a certain normal distribution, find the standard deviation ? when $\mu$ = 50 and 10.56% of the area lies to the right of 55.

Accepted Answer

The z-score corresponding to 10.56% in the right tail is approximately 1.25. Using the formula $ z = \frac{X - \mu}{\sigma} $, where $ X = 55 $, $ \mu = 50 $, and $ z = 1.25 $, we solve for $ \sigma $ to get $ \sigma = \frac{55 - 50}{1.25} = 4 $.

Question 3

A mathematics professor gives two different tests to two sections of his college algebra courses. The first class
has a mean of 56 with a standard deviation of 9 while the second class has a mean of 75 with a standard
deviation of 15. A student from the first class scores a 62 on the test while a student from the second class
scores an 83 on the test. Compare the scores.

Accepted Answer

The answer of A mathematics professor gives two different tests...

Question 4

Assume that the heights of women are normally distributed with a mean of 62.2 inches and a standard deviation of 2.3 inches. Find $\mathrm { Q } _ { 3 } ,$ the third quartile that separates the bottom 75% from the top 25%.

Accepted Answer

The third quartile, $Q_3$, corresponds to the 75th percentile of a distribution. In a normal distribution, the 75th percentile can be found by using the Z-score table or formula for the standard normal distribution. The Z-score for the 75th percentile is approximately 0.674. Using the formula $X = \mu + Z\sigma$, where $\mu$ is the mean, $\sigma$ is the standard deviation, and $Z$ is the Z-score, we calculate $X = 62.2 + 0.674(2.3) = 62.2 + 1.6 = 63.8$.

Question 5

SAT scores have a mean of 1026 and a standard deviation of 209. ACT scores have a mean of 20.8 and a standard deviation of 4.8. A student takes both tests while a junior and scores 1130 on the SAT and 25 on the
ACT. Compare the scores.

Accepted Answer

To compare the scores, we convert them to z-scores. For the SAT: $z = \frac{1130 - 1026}{209} \approx 0.497$. For the ACT: $z = \frac{25 - 20.8}{4.8} \approx 0.875$. A higher z-score indicates a score further above the mean, so the ACT score is better.

Question 6

Assume that the salaries of elementary school teachers in the United States are normally distributed with a mean of $37,000 and a standard deviation of $5000. What is the cutoff salary for teachers in the bottom 10%?

Accepted Answer

The cutoff salary for the bottom 10% can be found using the z-score for 10%, which is approximately -1.28. Using the formula: cutoff = mean + (z-score * standard deviation), we get $37,000 + (-1.28 * $5,000) = $30,600.

Question 7

Assume that blood pressure readings are normally distributed with $\mu$ = 114 and $\sigma$ = 8. A researcher wishes to
select people for a study but wants to exclude the top and bottom 10 percent. What would be the upper and
lower readings to qualify people to participate in the study?

Accepted Answer

The answer of Assume that blood pressure readings are normally...

Question 8

Compare the scores: a score of 88 on a test with a mean of 79 and a score of 78 on a test with a mean of 70.

Accepted Answer

To compare the scores effectively, additional information such as the standard deviation of the scores on each test is needed. Without knowing the distribution or variability of the scores, it's impossible to determine which score is relatively better.

Question 9

In a certain normal distribution, find the mean $\mu$ when ? = 5 and 5.48% of the area lies to the left of 78.

Accepted Answer

Given that 5.48% of the area lies to the left of 78 in a normal distribution, this corresponds to a Z-score that reflects the 5.48th percentile. Looking up this percentile in a standard normal distribution table or using a calculator, we find that it corresponds to a Z-score of approximately -1.6 (since it's in the lower tail of the distribution). The Z-score formula is $Z = \frac{X - \mu}{\sigma}$, where $X$ is the value in the distribution, $\mu$ is the mean, and $\sigma$ is the standard deviation. Rearranging this formula to solve for $\mu$, we get $\mu = X - Z\sigma$. Substituting the given values ($X = 78$, $Z = -1.6$, $\sigma = 5$), we find $\mu = 78 - (-1.6 \times 5) = 78 + 8 = 86$.

Question 10

The scores on a mathematics exam have a mean of 70 and a standard deviation of 5. Find the x-value that corresponds to the z-score 2.33.

Accepted Answer

We use the formula:
z = (x - mean) / standard deviation
2.33 = (x - 70) / 5
11.65 = x - 70
x = 81.65
Rounding to one decimal place gives x = 81.7. So the answer is choice A). We can use the formula:
z = (x - μ) / σ
where z = 2.33, μ = 70, σ = 5
Solving for x, we get:
x = zσ + μ
x = 2.33(5) + 70
x = 81.65
Rounding to one decimal place, we get:
x = 81.7
Therefore, the answer is A).

Question 11

Assume that the salaries of elementary school teachers in the United States are normally distributed with a
mean of $29,000 and a standard deviation of $2000. What is the cutoff salary for teachers in the top 10%?

Accepted Answer

The answer of Assume that the salaries of elementary school...

Question 12

Compare the scores: a score of 220 on a test with a mean of 200 and a standard deviation of 21 and a score of 90 on a test with a mean of 80 and a standard deviation of 8.

Accepted Answer

A score of 90 on a test with a mean of 80 and a standard deviation of 8 is 1.25 standard deviations above the mean (90-80)/8 = 1.25, while a score of 220 on a test with a mean of 200 and a standard deviation of 21 is only about 0.95 standard deviations above the mean (220-200)/21 = 0.95. Therefore, relative to their respective distributions, the score of 90 is better.

Question 13

Compare the scores: a score of 75 on a test with a mean of 65 and a standard deviation of 8 and a score of 75 on a test with a mean of 70 and a standard deviation of 4.

Accepted Answer

The answer of Compare the scores: a score of 75...

Question 14

SAT scores have a mean of 1026 and a standard deviation of 209. ACT scores have a mean of 20.8 and a standard deviation of 4.8. A student takes both tests while a junior and scores 860 on the SAT and 16 on the
ACT. Compare the scores.

Accepted Answer

The student's SAT score is (860 - 1026) / 209 = -0.79 standard deviations below the mean, while their ACT score is (16 - 20.8) / 4.8 = -1.00 standard deviations below the mean. Therefore, the student's SAT score was better.

Question 15

Assume that the heights of men are normally distributed with a mean of 69.4 inches and a standard deviation
of 1.3 inches. If the top 5 percent and bottom 5 percent are excluded for an experiment, what are the cutoff
heights to be eligible for this experiment? Round your answers to one decimal place.

Accepted Answer

The answer of Assume that the heights of men are...

Question 16

The times for completing one circuit of a bicycle course are normally distributed with a mean of 62.9 minutes
and a standard deviation of 6.1 minutes. An association wants to sponsor a race but will cut the bottom 25% of
riders. In a trial run, what should be the cutoff time?

Accepted Answer

The answer of The times for completing one circuit of...

Question 17

Two high school students took equivalent language tests, one in German and one in French. The student taking the German test, for which the mean was 66 and the standard deviation was 8, scored an 82, while the student
Taking the French test, for which the mean was 27 and the standard deviation was 5, scored a 35. Compare the
Scores.

Accepted Answer

The student taking the German test scored 1.5 standard deviations above the mean, while the student taking the French test scored 1.6 standard deviations above the mean. However, since the standard deviation for the German test was larger, scoring 1.5 standard deviations above the mean is a more impressive feat than scoring 1.6 standard deviations above the mean on the French test. Therefore, a score of 82 with a mean of 66 and a standard deviation of 8 is better. To compare the scores, we need to use z-scores, which tell us how many standard deviations above or below the mean a particular score is.

For the German test: 
z-score = (82 - 66) / 8 = 2

For the French test: 
z-score = (35 - 27) / 5 = 1.6

A z-score of 2 indicates that the score of 82 is two standard deviations above the mean for the German test, while a z-score of 1.6 indicates that the score of 35 is 1.6 standard deviations above the mean for the French test. Since a z-score of 2 is higher than a z-score of 1.6, we can conclude that the score of 82 on the German test is better than the score of 35 on the French test. Therefore, choice A is the best answer.

Question 18

In a certain normal distribution, 6.3% of the area lies to the left of 36 and 6.3% of the area lies to the right of 42.
Find the mean μ and the standard deviation $\sigma.$

Accepted Answer

The answer of In a certain normal distribution, 6.3% of...

Question 19

The body temperatures of adults are normally distributed with a mean of 98.6° F and a standard deviation of 0.95° F. What temperature represents the 95th percentile?

Accepted Answer

The 95th percentile is the temperature that 95% of the population is below. Using a z-score table, we can find that the z-score corresponding to the 95th percentile is 1.645. We can then use the formula z = (x - mu) / sigma to solve for x, the temperature that represents the 95th percentile:1.645 = (x - 98.6) / 0.95x = 98.6 + 1.645 * 0.95x = 100.16° F

Question 20

IQ test scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the x-score that corresponds to a z-score of 2.33.

Accepted Answer

To find the x-score corresponding to a z-score of 2.33, use the formula x = μ + zσ, where μ is the mean and σ is the standard deviation. Substituting the given values: x = 100 + 2.33(15) = 100 + 34.95 = 134.95.

Quiz 5: Normal Probability Distributions

IQ test scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the x-score that corresponds to a z-score of -1.645.

In a certain normal distribution, find the standard deviation ? when μ\muμ = 50 and 10.56% of the area lies to the right of 55.

Assume that the heights of women are normally distributed with a mean of 62.2 inches and a standard deviation of 2.3 inches. Find Q3,\mathrm { Q } _ { 3 } ,Q3​, the third quartile that separates the bottom 75% from the top 25%.

SAT scores have a mean of 1026 and a standard deviation of 209. ACT scores have a mean of 20.8 and a standard deviation of 4.8. A student takes both tests while a junior and scores 1130 on the SAT and 25 on the ACT. Compare the scores.

Assume that the salaries of elementary school teachers in the United States are normally distributed with a mean of $37,000 and a standard deviation of $5000. What is the cutoff salary for teachers in the bottom 10%?

Assume that blood pressure readings are normally distributed with μ\muμ = 114 and σ\sigmaσ = 8. A researcher wishes to select people for a study but wants to exclude the top and bottom 10 percent. What would be the upper and lower readings to qualify people to participate in the study?

Compare the scores: a score of 88 on a test with a mean of 79 and a score of 78 on a test with a mean of 70.

In a certain normal distribution, find the mean μ\muμ when ? = 5 and 5.48% of the area lies to the left of 78.

The scores on a mathematics exam have a mean of 70 and a standard deviation of 5. Find the x-value that corresponds to the z-score 2.33.

Assume that the salaries of elementary school teachers in the United States are normally distributed with a mean of $29,000 and a standard deviation of $2000. What is the cutoff salary for teachers in the top 10%?

Compare the scores: a score of 220 on a test with a mean of 200 and a standard deviation of 21 and a score of 90 on a test with a mean of 80 and a standard deviation of 8.

Compare the scores: a score of 75 on a test with a mean of 65 and a standard deviation of 8 and a score of 75 on a test with a mean of 70 and a standard deviation of 4.

SAT scores have a mean of 1026 and a standard deviation of 209. ACT scores have a mean of 20.8 and a standard deviation of 4.8. A student takes both tests while a junior and scores 860 on the SAT and 16 on the ACT. Compare the scores.

The times for completing one circuit of a bicycle course are normally distributed with a mean of 62.9 minutes and a standard deviation of 6.1 minutes. An association wants to sponsor a race but will cut the bottom 25% of riders. In a trial run, what should be the cutoff time?

In a certain normal distribution, 6.3% of the area lies to the left of 36 and 6.3% of the area lies to the right of 42. Find the mean μ and the standard deviation σ.\sigma.σ.

The body temperatures of adults are normally distributed with a mean of 98.6° F and a standard deviation of 0.95° F. What temperature represents the 95th percentile?

IQ test scores are normally distributed with a mean of 100 and a standard deviation of 15. Find the x-score that corresponds to a z-score of 2.33.

In a certain normal distribution, find the standard deviation ? when $\mu$ = 50 and 10.56% of the area lies to the right of 55.

Assume that the heights of women are normally distributed with a mean of 62.2 inches and a standard deviation of 2.3 inches. Find $\mathrm { Q } _ { 3 } ,$ the third quartile that separates the bottom 75% from the top 25%.

Assume that blood pressure readings are normally distributed with $\mu$ = 114 and $\sigma$ = 8. A researcher wishes to select people for a study but wants to exclude the top and bottom 10 percent. What would be the upper and lower readings to qualify people to participate in the study?

In a certain normal distribution, find the mean $\mu$ when ? = 5 and 5.48% of the area lies to the left of 78.

In a certain normal distribution, 6.3% of the area lies to the left of 36 and 6.3% of the area lies to the right of 42. Find the mean μ and the standard deviation $\sigma.$