Question 1

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°C at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some give readings below 0°C (denoted by negative numbers) and some give readings above 0°C (denoted by positive numbers). Assume that the mean reading is 0°C and the standard deviation of the readings is 1.00°C. Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. Find the temperature reading corresponding to the given information.
-Find $\mathrm { Q } _ { 3 }$, the third quartile.
A) $- 1.3 ^ { \circ }$
B) $0.67 ^ { \circ }$
C) $0.53 ^ { \circ }$
D) $0.82 ^ { \circ }$

Accepted Answer

The answer of The Precision Scientific Instrument Company manufactures thermometers...

Question 2

Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.
-
A) 0.8485
B) 0.8708
C) 0.1292
D) 0.8907

Accepted Answer

The answer of Find the area of the shaded region....

Question 3

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°C at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing point of water, some give readings below 0°C (denoted by negative numbers) and some give readings above 0°C (denoted by positive numbers). Assume that the mean reading is 0°C and the standard deviation of the readings is 1.00°C. Also assume that the frequency distribution of errors closely resembles the normal distribution. A thermometer is randomly selected and tested. Find the temperature reading corresponding to the given information.
-A quality control analyst wants to examine thermometers that give readings in the bottom 7%. Find the reading that separates the bottom 7% from the others. 
A) $- 1.63 ^ { \circ }$
B) $- 1.75 ^ { \circ }$
C) $- 1.89 ^ { \circ }$
D) $- 1.48 ^ { \circ }$

Accepted Answer

The answer of The Precision Scientific Instrument Company manufactures thermometers...

Question 4

Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.
-Shaded area is 0.0694. 
A) 1.48
B) 1.39
C) 1.45
D) 1.26

Accepted Answer

The answer of Find the indicated z score. The graph...

Question 5

If z is a standard normal variable, find the probability.
-$\mathrm { P } ( \mathrm { z } > 0.59 )$

A) 0.7224
B) 0.2776
C) 0.2224
D) 0.2190

Accepted Answer

The answer of If z is a standard normal variable,...

Question 6

Scores on a test have a mean of 71 and $Q _ { 3 }$ is 82 . The scores have a distribution that is  approximately normal. Find P90. (You will need to first find the standard deviation.)
A) 90.4
B) 92.7
C) 91
D) 92

Accepted Answer

To find P90, we first need to find the standard deviation. In a normal distribution, Q3 corresponds to the 75th percentile. Using the Z-score for the 75th percentile (approximately 0.675), we can calculate the standard deviation: $ \sigma = \frac{82 - 71}{0.675} \approx 16.3 $. Then, using the Z-score for the 90th percentile (approximately 1.28), we find P90: $ P90 = 71 + 1.28 \times 16.3 \approx 92 $.

Question 7

Computers are often used to randomly generate digits of telephone numbers to be called for telemarketing purposes. Can the methods used with normal probability distributions be used to find the probability that when one digit is randomly generated, it is greater than 5? Why or why not? What is the probability of getting a digit greater than 5?
A) Yes, because the generated digits have a normal distribution. The probability of a digit greater than 5 is 0.4.
B) No, because the generated digits do not have a normal distribution. The probability of a digit greater than 5 is 0.6.
C) Yes, because the generated digits have a normal distribution. The probability of a digit greater than 5 is 0.6.
D) No, because the generated digits do not have a normal distribution. The probability of a digit greater than 5 is 0.4.

Accepted Answer

The answer of Computers are often used to randomly generate...

Question 8

Find the area of the shaded region. The graph depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). 
A) 0.7303
B) 0.7619
C) 0.7938
D) 0.7745

Accepted Answer

The answer of Find the area of the shaded region....

Question 9

Use the normal distribution to approximate the desired probability.
-A coin is tossed 20 times. A person, who claims to have extrasensory perception, is asked to predict the outcome of each flip in advance. She predicts correctly on 14 tosses. What is the probability of being correct 14 or more times by guessing? Does this probability seem to verify her claim?
A) 0.0582, no
B) 0.4418, yes
C) 0.0582, yes
D) 0.4418, no

Accepted Answer

The probability of guessing correctly 14 or more times out of 20 can be approximated using the normal distribution. The mean is 10 and the standard deviation is approximately 2.24. The z-score for 14 is about 1.79, which corresponds to a probability of about 0.0582. This low probability suggests that her claim might be valid.

Question 10

Use the normal distribution to approximate the desired probability.
-A coin is tossed 20 times. A person, who claims to have extrasensory perception, is asked to predict the outcome of each flip in advance. She predicts correctly on 16 tosses. What is the probability of being correct 16 or more times by guessing? Does this probability seem to verify her claim?
A) 0.4931, yes
B) 0.0069, yes
C) 0.4931, no
D) 0.0069, no

Accepted Answer

The answer of Use the normal distribution to approximate the...

Question 11

A math teacher gives two different tests to measure students' aptitude for math. Scores on the first test are normally distributed with a mean of 23 and a standard deviation of 5.3. Scores on the second test are normally distributed with a mean of 71 and a standard deviation of 11.9. Assume that the two tests use different scales to measure the same aptitude. If a student scores 29 on the first test, what would be his equivalent score on the second test? (That is, find the score that would put him in the same percentile.)
A) 84
B) 85
C) 77
D) 82

Accepted Answer

The z-score for the first test is (29-23)/5.3 = 1.13. The corresponding percentile is 86.8%. The z-score for the second test that corresponds to the 86.8% percentile is (x-71)/11.9 = 1.13. Solving for x gives x = 84.

Question 12

Assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of pounds lost.
-Less than 11 pounds 
A) $\frac { 5 } { 7 }$

B) $\frac { 1 } { 6 }$

C) $\frac { 5 } { 6 }$

D) $\frac { 1 } { 3 }$

Accepted Answer

The range of weight loss is from 6 to 12 pounds. The probability of losing less than 11 pounds is the length of the interval from 6 to 11 divided by the total length of the interval from 6 to 12, which is $\frac{11-6}{12-6} = \frac{5}{6}$.

Question 13

Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.
-Shaded area is 0.0401. 
A) $- 1.75$
B) $- 1.63$
C) $- 1.48$
D) $- 1.89$

Accepted Answer

The answer of Find the indicated z score. The graph...

Question 14

Assume that X has a normal distribution, and find the indicated probability.
-The mean is $\mu = 60.0$ and the standard deviation is $\sigma = 4.0$. 
Find the probability that $X$ is less than $53.0$.
A) 0.0401
B) 0.9599
C) 0.5589
D) 0.0802

Accepted Answer

To find the probability that $X < 53.0$, we first calculate the z-score: $z = \frac{53 - 60}{4} = -1.75$. Using the standard normal distribution table, the probability corresponding to $z = -1.75$ is approximately 0.0401.

Question 15

If z is a standard normal variable, find the probability.
-$\mathrm { P } ( - 0.73 < \mathrm { z } < 2.27 )$

A) 0.2211
B) 0.7557
C) 1.54
D) 0.4884

Accepted Answer

The probability that a standard normal variable z falls between -0.73 and 2.27 can be found using the standard normal distribution table or a calculator. The cumulative probability for z = 2.27 is approximately 0.9884, and for z = -0.73 is approximately 0.2327. The probability is 0.9884 - 0.2327 = 0.7557.

Question 16

Use the normal distribution to approximate the desired probability.
-Find the probability that in 200 tosses of a fair die, we will obtain at most 30 fives.
A) 0.1871
B) 0.3229
C) 0.2946
D) 0.4936

Accepted Answer

The answer of Use the normal distribution to approximate the...

Question 17

Suppose that replacement times for washing machines are normally distributed with a mean of 10.9 years and a standard deviation of 1 years. Find the replacement time that separates the top 18% from the bottom 82%.
A) 11.1 years
B) 10.0 years
C) 11.0 years
D) 11.8 years

Accepted Answer

To find the replacement time that separates the top 18% from the bottom 82%, we need to find the z-score that corresponds to the 82nd percentile of a standard normal distribution. This z-score is approximately 0.915. Using the formula X = μ + zσ, where μ = 10.9 and σ = 1, we calculate X = 10.9 + 0.915(1) = 11.815, which rounds to 11.8 years.

Question 18

A final exam in Math 160 has a mean of 73 with standard deviation 7.8. If 24 students are randomly selected, find the probability that the mean of their test scores is less than 70.
A) 0.1006
B) 0.0278
C) 0.9699
D) 0.0301

Accepted Answer

The answer of A final exam in Math 160 has...

Question 19

A study of the amount of time it takes a mechanic to rebuild the transmission for a 2005 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanicsare randomly selected, find the probability that their mean rebuild time exceeds 9.1 hours.
A) 0.1046
B) 0.0069
C) 0.0046
D) 0.1285

Accepted Answer

The probability can be found using the standard normal distribution. First, calculate the standard error (SE) using the formula SE = σ/√n = 1.8/√40. Then, find the z-score for 9.1 hours using the formula z = (X̄ - μ) / SE. Finally, use the z-score to find the probability from the standard normal distribution table. The probability that the mean exceeds 9.1 hours corresponds to choice B.

Question 20

The incomes of trainees at a local mill are normally distributed with a mean of $1100 and a standard deviation of $150. What percentage of trainees earn less than $900 a month?
A) 9.18%
B) 40.82%
C) 35.31%
D) 90.82%

Accepted Answer

To find the percentage of trainees earning less than $900, we calculate the z-score: $z = \frac{X - \mu}{\sigma} = \frac{900 - 1100}{150} = -\frac{200}{150} = -1.33$. Looking up -1.33 on the standard normal distribution table gives us approximately 9.18%.

Quiz 6: Normal Probability Distributions

Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. -

Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. -Shaded area is 0.0694.

If z is a standard normal variable, find the probability. - $\mathrm { P } ( \mathrm { z } > 0.59 )$

Scores on a test have a mean of 71 and $Q _ { 3 }$ is 82 . The scores have a distribution that is approximately normal. Find P90. (You will need to first find the standard deviation.)

Find the area of the shaded region. The graph depicts IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test) .

Assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of pounds lost. -Less than 11 pounds

Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. -Shaded area is 0.0401.

Assume that X has a normal distribution, and find the indicated probability. -The mean is $\mu = 60.0$ and the standard deviation is $\sigma = 4.0$ . Find the probability that $X$ is less than $53.0$ .

If z is a standard normal variable, find the probability. - $\mathrm { P } ( - 0.73 < \mathrm { z } < 2.27 )$

Use the normal distribution to approximate the desired probability. -Find the probability that in 200 tosses of a fair die, we will obtain at most 30 fives.

Suppose that replacement times for washing machines are normally distributed with a mean of 10.9 years and a standard deviation of 1 years. Find the replacement time that separates the top 18% from the bottom 82%.

A final exam in Math 160 has a mean of 73 with standard deviation 7.8. If 24 students are randomly selected, find the probability that the mean of their test scores is less than 70.

A study of the amount of time it takes a mechanic to rebuild the transmission for a 2005 Chevrolet Cavalier shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanicsare randomly selected, find the probability that their mean rebuild time exceeds 9.1 hours.

The incomes of trainees at a local mill are normally distributed with a mean of $1100 and a standard deviation of $150. What percentage of trainees earn less than $900 a month?