Question 1

A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. Find P60, the score which separates the lower 60% from the top
40%)

Accepted Answer

To find P60, we need to find the score that separates the lower 60% from the top 40%. This means that the area under the normal curve to the left of P60 is 0.60 and the area to the right of P60 is 0.40.
Using a standard normal distribution table, we find the z-score that corresponds to a cumulative area of 0.60 is approximately 0.25. 
Now we can use the formula: 
z = (x - μ) / σ 
where z is the z-score, x is the score we want to find (P60), μ is the mean (200), and σ is the standard deviation (50).

Plugging in the values we get:
0.25 = (P60 - 200) / 50

Solving for P60, we get:
P60 = 212.5

Therefore, the answer is C) 212.5.

Question 2

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 mechanics are randomly selected, find the probability that their mean rebuild time is less than 8.9 hours.

Accepted Answer

The central limit theorem states that the distribution of sample means will be approximately normal, regardless of the shape of the population distribution, if the sample size is large enough. In this case, the sample size is 40, which is large enough to apply the central limit theorem.The mean of the sample means will be equal to the mean of the population, which is 8.4 hours. The standard deviation of the sample means will be equal to the standard deviation of the population divided by the square root of the sample size, which is 1.8 hours / sqrt(40) = 0.28 hours.Therefore, the probability that the mean rebuild time of the 40 mechanics is less than 8.9 hours is equal to the probability that a normally distributed random variable with a mean of 8.4 hours and a standard deviation of 0.28 hours is less than 8.9 hours. This probability can be calculated using a standard normal distribution table or a calculator, and it is equal to 0.9608.

Question 3

Construct a normal quartile plot of the given data. Use your plot to determine whether the data come from a normally distributed population.
-The systolic blood pressure (in mmHg) is given below for a sample of 12 men aged between 60 and 65. 
$$\begin{array} { l l l l } 
127 & 135 & 118 & 164 \
143 & 130 & 125 & 153 \
120 & 173 & 140 & 180
\end{array}$$

Accepted Answer

The answer of Construct a normal quartile plot of the...

Question 4

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 P₄₀, the 40th percentile.

Accepted Answer

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

Question 5

In one region, the September energy consumption levels for single-family homes are found to be normally distributed with a mean of 1050 kWh and a standard deviation of 218 kWh. If 50 different homes are randomly selected, find the probability that their mean energy consumption level for September is greater than 1075 kWh.

Accepted Answer

The answer of In one region, the September energy consumption...

Question 6

Under what conditions are you allowed to use the normal distribution to approximate the binomial distribution? Under what conditions might you want to use the normal distribution to approximate the binomial as opposed to using the binomial probability formula, a table of binomial probabilities, or a computer?

Accepted Answer

The answer of Under what conditions are you allowed to...

Question 7

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 11 tosses. What is the probability of being correct 11 or more times by guessing? Does this probability seem to verify her claim?

Accepted Answer

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

Question 8

Assume that z scores are normally distributed with a mean of 0 and a standard deviation of 1. If P(z > c) = 0.1093, find c.

Accepted Answer

The z-score corresponding to the probability P(z > c) = 0.1093 is approximately 1.23, as this is the value where the cumulative probability from the right tail is 0.1093.

Question 9

Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). Find the probability that a randomly selected adult has an IQ between 90 and 120 (somewhere in the range of normal to bright normal).

Accepted Answer

The answer of Assume that adults have IQ scores that...

Question 10

Suppose that replacement times for washing machines are normally distributed with a mean of 9.3 years and a standard deviation of 1.1 years. Find the probability that 70 randomly selected washing machines will have a mean replacement time less than 9.1 years.

Accepted Answer

The standard error of the mean is 1.1/sqrt(70). The z-score for a mean of 9.1 is (9.1 - 9.3) / (1.1/sqrt(70)). Using the z-score, we find the probability from the standard normal distribution table.

Question 11

Human body temperatures are normally distributed with a mean of 98.20°F and a standard deviation of 0.62°F. Find the temperature that separates the top 7% from the bottom 93%. Round to the nearest hundredth of a degree.

Accepted Answer

The z-score for the top 7% is 1.41 (from a z-table). So, the temperature that separates the top 7% from the bottom 93% is 98.20 + 1.41 * 0.62 = 99.12°F.

Question 12

Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.
-Shaded area is 0.8599.

Accepted Answer

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

Question 13

Sketch a brief diagram of the standard normal distribution table. You only need to show two sets of values. Identify the z scores and the area scores in the table by circling the scores and writing z score and area by each one. Describe how to find the area corresponding to a given z score. Describe how to find the the z score corresponding to a given area value.

Accepted Answer

The answer of Sketch a brief diagram of the standard...

Question 14

The volumes of soda in quart soda bottles are normally distributed with a mean of 32.3 oz and a standard deviation of 1.2 oz. What is the probability that the volume of soda in a randomly selected bottle will be less than 32 oz?

Accepted Answer

The answer of The volumes of soda in quart soda...

Question 15

Find the indicated IQ score. 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).  The shaded area under the curve is 0.10.

Accepted Answer

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

Question 16

The ages (in years) of the four U.S. vice presidents who assumed office after presidential assassinations are 56 (A. Johnson), 51 (C. Arthur), 42 (T. Roosevelt), and 55 (L.B. Johnson).
(i) Assuming that 2 of the ages are randomly selected with replacement, list the 16 different possible samples. 
(ii) Find the median of each of the 16 samples, then summarize the sampling distribution of the medians in the format of a table representing the probability distribution. 
(iii) Compare the population median to the mean of the sample medians. 
(iv) Do the sample medians target the value of the population median? In general, do sample medians make good estimators of population medians? Why or why not?

Accepted Answer

The answer of The ages (in years) of the four...

Question 17

A normal quartile plot is given below for a sample of scores on an aptitude test. Use the plot to assess the normality of scores on this test. Explain your reasoning.

Accepted Answer

The answer of A normal quartile plot is given below...

Question 18

In one region, the September energy consumption levels for single-family homes are found to be normally distributed with a mean of 1050 kWh and a standard deviation of 218 kWh. For a randomly selected home, find the probability that the September energy consumption level is between 1100 kWh and 1225 kWh.

Accepted Answer

The answer of In one region, the September energy consumption...

Question 19

The given values are discrete. Use the continuity correction and describe the region of the normal distribution that corresponds to the indicated probability.
-The probability of more than 56 correct answers

Accepted Answer

The continuity correction adds 0.5 to the upper and lower bounds of the interval, so the region of the normal distribution that corresponds to the probability of more than 56 correct answers is the area to the right of 56.5.

Question 20

Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later. The numbers of people in the households are 5, 6, and 10. Consider the values of 5, 6, and 10 to be a population. Assume that samples of size n = 2 are randomly selected with replacement from the population of 5, 6, and 10. The nine different samples are as follows:
(5, 5), (5, 6), (5, 10), (6, 5), (6, 6), (6, 10), (10, 5), (10, 6), and (10, 10). 
(i) Find the variance of each of the nine samples, then summarize the sampling distribution of the variances in the format of a table representing the probability distribution. 
(ii) Compare the population variance to the mean of the sample variances. 
(iii) Do the sample variances target the value of the population variance? In general, do variances make good estimators of population variances? Why or why not?

Accepted Answer

The answer of Three randomly selected households are surveyed as...

Quiz 6: Normal Probability Distributions

A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. Find P60, the score which separates the lower 60% from the top 40%)

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 mechanics are randomly selected, find the probability that their mean rebuild time is less than 8.9 hours.

Assume that z scores are normally distributed with a mean of 0 and a standard deviation of 1. If P(z > c) = 0.1093, find c.

Assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test) . Find the probability that a randomly selected adult has an IQ between 90 and 120 (somewhere in the range of normal to bright normal) .

Suppose that replacement times for washing machines are normally distributed with a mean of 9.3 years and a standard deviation of 1.1 years. Find the probability that 70 randomly selected washing machines will have a mean replacement time less than 9.1 years.

Human body temperatures are normally distributed with a mean of 98.20°F and a standard deviation of 0.62°F. Find the temperature that separates the top 7% from the bottom 93%. Round to the nearest hundredth of a degree.

Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. -Shaded area is 0.8599.

The volumes of soda in quart soda bottles are normally distributed with a mean of 32.3 oz and a standard deviation of 1.2 oz. What is the probability that the volume of soda in a randomly selected bottle will be less than 32 oz?

Find the indicated IQ score. 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) . The shaded area under the curve is 0.10.

A normal quartile plot is given below for a sample of scores on an aptitude test. Use the plot to assess the normality of scores on this test. Explain your reasoning.

The given values are discrete. Use the continuity correction and describe the region of the normal distribution that corresponds to the indicated probability. -The probability of more than 56 correct answers