# EBook Problems Normal Std

## EBook Problems Set - The Standard Normal Distribution and Central Limit Theorem Problems

These problems demonstrate concepts in 2 sections of the EBook -- The Standard Normal Distribution and The Central Limit Theorem. Problems that ask for the distribution (or probabilities) of a single random number, or a sample-average of random numbers, pertain to section on Standard Normal Distribution, or section on Central Limit Theorem, respectively.

### Problem 1

Weight is a measure that tends to be normally distributed. Suppose the mean weight of all women at a large university is 135 pounds, with a standard deviation of 12 pounds. If you were to randomly sample 9 women at the university, there would be a 68% chance that the sample mean weight would be between:

(a) 131 and 139 pounds.
(b) 133 and 137 pounds.
(c) 119 and 151 pounds
(d) 125 and 145 pounds.
(e) 123 and 147 pounds.

### Problem 2

The amount of money college students spend each semester on textbooks is normally distributed with a mean of \$195 and a standard deviation of \$20. Suppose you take a random sample of 100 college students from this population. There is a 68% chance that the sample mean amount spent on textbooks is between:

(a) \$193 and \$197.
(b) \$155 and \$235.
(c) \$191 and \$199.
(d) \$175 and \$215.

### Problem 3

A researcher converts 100 lung capacity measurements to z-scores. The lung capacity measurements do not follow a normal distribution. What can we say about the standard deviation of the 100 z-scores?

(a) It depends on the standard deviation of the raw scores
(b) It equals 1
(c) It equals 100
(d) It must always be less than the standard deviation of the raw scores
(e) It depends on the shape of the raw score distribution

### Problem 4

The weights of packets of cookies produced by a certain manufacturer have a normal distribution with a mean of 202 grams and a standard deviation of 3 grams. What is the weight that should be stamped on the packet so that only 0.99% of packets are underweight?

(a) 200
(b) 195
(c) 190
(d) 205

### Problem 5

GSP Inc. is trying two different marketing techniques for its toothpaste. In 20 test cities, it is using family branding. This sells toothpaste with a mean of 2,250 units per week and a standard deviation of 250 units per week. In 20 other test cities, GSP is using individual branding. This sells toothpaste with a mean of 2,250 units per week and a standard deviation of 500 units per week. GSP wants to select the marketing technique that sells at least 2,350 units per week more often. If the number of units sold per week follows a normal distribution, which marketing technique should GSP choose?

(a) Individual Branding
(b) Can't be answered with the information given
(c) Family Branding
(d) They each get the same result

### Problem 6

Among first year students at a certain university, scores on the verbal SAT follow the normal curve. The average is around 500 and the SD is about 100. Tatiana took the SAT, and placed at the 85% percentile. What was her verbal SAT score?

(a) 604
(b) 560
(c) 90
(d) 403

### Problem 7

A set of test scores are normally distributed. The mean is 100 and the standard deviation is 20. These scores are converted to z-scores. What are the z-scores of the mean and median?

(a) 1
(b) 100
(c) 0
(d) 50

### Problem 8

In Japan there is an annual turkey dog eating contest. The number of turkey dogs that contestants eat are normally distributed with a mean of 36 turkey dogs and a standard deviation of 6 turkey dogs. A contestant eats 27 turkey dogs. What is his z-score?

(a) 6
(b) -1.5
(c) 9
(d) 1.5
(e) -9

### Problem 9

Sauron the Dark Lord of Mordor, when not busy trying to take over Middle Earth or searching for his lost Ring of Power, likes to indulge in statistics. One day he decided to estimate the average weight of his orc soldiers, which he knows to be normally distributed.

Sauron took a random sample 100 orc soldiers and found the mean and the standard deviation to be 200lbs and and 20lbs respectively. He can be 68% confident that the mean weight in the population of orc soldiers is between

(a) 198 to 202 lbs
(b) 194 to 206 lbs
(c) None of the above
(d) 196 to 204 lbs

### Problem 10

Years ago, the value of HBA1c, a test used to measure blood sugar level, was normally distributed with mean 6 and standard deviation 1. A diabetic person is anyone whose HBA1c is larger than 7. We want to find out (a) If I choose a person at random from the population, what is the probability that this person is NOT a diabetic? (b) If I take a random sample of 5 people what is the probability that their average HBAic is smaller than 7?

(a) (a) approximately 0.9772 (b) approximately 0.0228
(b) (a) approximately 0.8413 (b) approximately 1
(c) None of the above
(d) (a) approximately 0.8413 (b) approximately 0.9871

### Problem 11

Fluorescent light bulbs have lifetimes that follow a normal distribution, with an average life of 1,685 days and a standard deviation of 1,356 hours. In the production process the manufacturer draws random samples of 197 lightbulbs and determines the mean lifetime of the sample. What is the standard deviation of this sample mean?

(a) 1356
(b) 8.553
(c) 6.883
(d) 96.611

### Problem 12

The Rockwell hardness of certain metal pins is known to have a mean of 50 and a standard deviation of 1.5. If the distribution of all such pin hardness measurements is known to be normal, what is the probability that the average hardness for a random sample of nine pins is at least 50.5?

(a) Approximately 4
(b) 0.4
(c) Approximately 0.1587
(d) Approximately 0

### Problem 13

```If we draw the Normal probability plot for the following histogram, what will it show and how will it appear?
```
(a) It will show the expected cumulative percentiles vs. the actual cumulative percentiles and the points will deviate from a straight diagonal line.
(b) It will show the expected cumulative percentiles vs. the actual cumulative percentiles and all the points will approximate a diagonal straight line.
(c) It will show a histogram of the Z scores and it will be positively skewed.
(d) It will show a histogram of the Z scores and it will be normal.

### Problem 14

The weights of packets of cookies produced by a certain manufacturer have a normal distribution with a mean of 202 grams and a standard deviation of 3 grams. What is the weight that should be stamped on the packet so that only 0.99% of packets are underweight?

(a) 190
(b) 195
(c) 205
(d) 200

### Problem 15

In a large lecture course, the scores on the final examination followed the normal curve closely. The average score was 60 points and one fourth of the class scored between 50 and 70 points. The SD of the scores was

(a) A larger than 10 points
(b) 10 points
(c) Can't be determined with the information given
(d) Smaller than 10 points

### Problem 16

Given that the IQ scores in the population follow the normal distribution with mean (μ) equal to 100 and standard deviation (σ) equal to 15, what is the best answer?

(a) If you pick a person at random, the chance that his IQ falls between 115-130 is more than the chance that his IQ falls between 60-85.
(b) If you pick a person at random, the chance that his IQ falls between 115-130 is equal to the chance that his IQ falls between 60-85.
(c) If you pick a person at random, the chance that his IQ falls between 115-130 is not comparable to his IQ falling between 60-85.
(d) If you pick a person at random, the chance that his IQ falls between 115-130 is less than the chance that his IQ falls between 60-85.

### Problem 17

Scott's percentile rank in the verbal section of the SAT was 80. What can be assumed about his score?

(a) Scott got 80% of the questions right
(b) 80% of the students that took the test received a lower score than Scott
(c) 80% of the students that took the test scored higher than Scott did
(d) Scott answered at least 80% of the questions correctly

### Problem 18

GSP Inc. is trying two different marketing techniques for its toothpaste. In 20 test cities, it is using family branding. This sells toothpaste with a mean of 2,250 units per week and a standard deviation of 250 units per week. In 20 other test cities, GSP is using individual branding. This sells toothpaste with a mean of 2,250 units per week and a standard deviation of 500 units per week. GSP wants to select the marketing technique that sells at least 2,350 units per week more often. If the number of units sold per week follows a normal distribution, which marketing technique should GSP choose?

(a) They each get the same result
(b) Individual branding
(c) Family branding
(d) Can't be answered with the information given

### Problem 19

We read that the average weight for babies born in the United States is 7.5 pounds with deviation of 0.25 pounds. We can assume that the distribution of birth weights is nearly normal. If we select one baby at random, what are the chances that the baby weights less that 8 pounds?

(a) 0.9772
(b) 0.0456
(c) 0.9544
(d) 0.0228

### Problem 20

In an article in the Journal of American Pediatric Health researchers claim that the weights of healthy babies born in the United States form a distribution that is nearly normal with an average weight of 7.25 pounds and standard deviation of 1.75 pounds. The US Department of Health classifies a newborn as "low birth weight" if her/his weight is less than 5.5 pounds. What is the probability that a baby, chosen at random, weighs less than 5.5 pounds?

(d) the probability cannot be determined

### Problem 21

Exam Times Suppose that the distribution of time required for a college student to complete a standardized exam is nearly normal with mean 45 minutes and standard deviation of 5 minutes. If you select 3 students at random, what are the chances that all 3 students finish in less than 35 minutes?