Deck 10: The Normal Distribution

Question 1

The total area under the normal probability distribution is equal to which of the following?

A) 0
B) 0.5
C) 1.0
D) This value depends on the mean and variance.

Answer 1

C

Question 2

For the equation for a normal distribution shown, what are the mean and variance?
?
$f(x)=\frac{1}{\sqrt{32 \pi}} e^{\frac{-(x-16)^{2}}{32}}$

A) Mean = 4, variance = 4
B) Mean = 4, variance = 16
C) Mean = 16, variance = 4
D) Mean = 16, variance = 16

Answer 2

Mean = 16, variance = 16

Question 3

For the equation for a normal distribution shown, what are the mean and standard deviation?
?
$f(x)=\frac{1}{\sqrt{32 \pi}} e^{\frac{-(x-40)^{2}}{32}}$

A) Mean = 40, variance = 4
B) Mean = 40, variance = 8
C) Mean = 40, variance = 16
D) Mean = 40, variance = 24

Answer 3

Mean = 40, variance = 16

Question 4

Which equation shown is correct for a normal distribution with a mean of 20 and a variance of 10?

A) $f(x)=\frac{1}{\sqrt{20 \pi}} e^{\frac{-(x-10)^{2}}{200}}$
B) $f(x)=\frac{1}{\sqrt{20 \pi}} e^{\frac{-(x-20)^{2}}{200}}$
C) $f(x)=\frac{1}{\sqrt{200 \pi}} e^{\frac{-(x-10)^{2}}{32}}$
D) $f(x)=\frac{1}{\sqrt{200} \pi} e^{\frac{-(x-20)^{2}}{32}}$

Question 5

Which equation shown is correct for a normal distribution with a median of 25 and a standard deviation of 5?

A) $f(x)=\frac{1}{\sqrt{10 \pi}} e^{\frac{-(x-25)^{2}}{25}}$
B) $f(x)=\frac{1}{\sqrt{25 \pi}} e^{\frac{-(x-25)^{2}}{25}}$
C) $f(x)=\frac{1}{\sqrt{50 \pi}} e^{\frac{-(x-25)^{2}}{50}}$
D) $f(x)=\frac{1}{\sqrt{100 \pi}} e^{\frac{-(x-100)^{2}}{50}}$

Question 6

Which of the following is not a property of the normal distribution?

A) It is a continuous distribution.
B) It is a discrete distribution.
C) It is a symmetric distribution.
D) Its mean and mode are equal.

Question 7

Which of the following is not a good rule of thumb for data values exhibiting a normal distribution?

A) Approximately 95% of the values lie within two standard deviations of the mean.
B) Approximately two-thirds of the values lie within one standard deviation of the mean.
C) Exactly half of the values lie at or above the mean.
D) Exactly half of the values lie within one standard deviation of the mean.

Question 8

For values from a data set, when we take the value, subtract the mean, and then divide by the standard deviation, we create new values called which of the following??

A) Scaled normal deviate
B) Scaled normal value
C) Standard normal deviate
D) Standard normal value

Question 9

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. To the nearest integer, how many values are less than 93?

A) 562
B) 582
C) 622
D) 692

Question 10

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. To the nearest integer, how many values are greater than 84?

A) 757
B) 769
C) 785
D) 791

Question 11

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. To the nearest integer, how many values are between 87 and 99?

A) 278
B) 562
C) 703
D) 840

Question 12

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. What value of Z below most closely corresponds to the 65th percentile?

A) 91.7
B) 91.9
C) 92.1
D) 92.3

Question 13

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. What value below is closest to the value larger than exactly 55% of the population?

A) 90.6
B) 90.8
C) 91.0
D) 91.2

Question 14

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. What proportion of values are larger than 78?

A) 0.0228
B) 0.8666
C) 0.9772
D) 0.9975

Question 15

Consider a population of 900 values that are normally distributed with a mean of 90 and standard deviation of 6. Which value below is closest to the IQR?

A) 8.1
B) 8.3
C) 8.5
D) 8.7

Question 16

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. To the closest integer, how many values in the population are less than 62?

A) 599
B) 898
C) 1,103
D) 1,314

Question 17

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. To the closest integer, how many values in the population are greater than 54?

A) 1,140
B) 1,150
C) 1,160
D) 1,170

Question 18

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. To the closest integer, how many values in the population are between 58 and 65?

A) 502
B) 509
C) 519
D) 529

Question 19

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. The 75th percentile most closely corresponds to which value of Z?

A) 65.0
B) 65.2
C) 65.4
D) 65.6

Question 20

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. What population value is larger than exactly 40% of the population?

A) 57.06
B) 57.36
C) 57.66
D) 57.96

Question 21

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. What proportion of values in the population are larger than 63?

A) 0.334
B) 0.354
C) 0.374
D) 0.394

Question 22

Imagine a population consisting of 1,500 values that are normally distributed around a mean of 60, which exhibits a standard deviation of 8. Which value below is closest to the IQR for the population?

A) 8.0
B) 8.6
C) 10.2
D) 10.8

Question 23

For a set of samples taken from a normally distributed population, which of the following is not generally true?

A) The distribution of sample means is centered around the population mean.
B) The distribution of sample means is normal.
C) The distribution of sample means is wider for larger population variances.
D) The distribution of sample means is wider for larger sample sizes.

Question 24

Crossbill finches have beaks that overlap with the top beak to one side or the other of the lower beak. Most populations have 50:50 ratios of "right-handed" and "left-handed" beaks, and the handedness of the beaks is thought to be randomly determined, so we expect to see equal numbers of right-beak and left-beak finches. Consider a survey of 140 crossbill finches, which reveals 82 with right beaks and 58 with left beaks. What is the P-value of a two-sided binomial test using the normal distribution to approximate the binomial?

A) 0.026
B) 0.052
C) 0.078
D) 0.104

Question 25

Crossbill finches have beaks that overlap with the top beak to one side or the other of the lower beak. Most populations have 50:50 ratios of "right-handed" and "left-handed" beaks, and the handedness of the beaks is thought to be randomly determined, so we expect to see equal numbers of right-beak and left-beak finches. Consider a survey of 140 crossbill finches, which reveals 82 with right beaks and 58 with left beaks. What is the conclusion of a two-sided binomial test using the normal distribution to approximate the binomial?

A) Fail to reject null hypothesis, we lack evidence that the handedness of the beaks is non-random.
B) Fail to reject null hypothesis, we have evidence that the handedness of the beaks is non-random.
C) Reject null hypothesis, we lack evidence that the handedness of the beaks is non-random.
D) Reject null hypothesis, we have evidence that the handedness of the beaks is non-random.

Question 26

Crossbill finches have beaks that overlap with the top beak to one side or the other of the lower beak. Most populations have 50:50 ratios of "right-handed" and "left-handed" beaks, and the handedness of the beaks is thought to be randomly determined, so we expect to see equal numbers of right-handed and left-handed finches. Consider a survey of 140 crossbill finches, which reveals 84 with right beaks and 56 with left beaks. What is the P-value of a two-sided binomial test using the normal distribution to approximate the binomial?

A) 0.015
B) 0.020
C) 0.023
D) 0.033

Question 27

Crossbill finches have beaks that overlap with the top beak to one side or the other of the lower beak. Most populations have 50:50 ratios of "right-handed" and "left-handed" beaks, and the handedness of the beaks is thought to be randomly determined, so we expect to see equal numbers of right-beak and left-beak finches. Consider a survey of 140 crossbill finches, which reveals 84 with right beaks and 56 with left beaks. What is the conclusion of a two-sided binomial test using the normal distribution to approximate the binomial?

A) Fail to reject null hypothesis, we lack evidence that the handedness of the beaks is non-random.
B) Fail to reject null hypothesis, we have evidence that the handedness of the beaks is non-random.
C) Reject null hypothesis, we lack evidence that the handedness of the beaks is non-random.
D) Reject null hypothesis, we have evidence that the handedness of the beaks is non-random.

Question 28

When mating two heterozygotes for alleles in which one is dominant to the other, if the usual Mendelian segregation process is occurring, the ratio of the offspring phenotypes produced should be 3:1. The dominant phenotype would be more common than the recessive one. Imagine a situation in which two heterozygotes are mated and among their 200 offspring, 160 show the dominant phenotype while 40 show the recessive phenotype. What is the P-value of a two-sided binomial test using the normal distribution to approximate the binomial?

A) 0.015
B) 0.031
C) 0.060
D) 0.121

Question 29

When mating two heterozygotes for alleles in which one is dominant to the other, if the usual Mendelian segregation process is occurring, the ratio of the offspring phenotypes produced should be 3:1. The dominant phenotype would be more common than the recessive one. Imagine a situation in which two heterozygotes are mated and among their 200 offspring, 160 show the dominant phenotype while 40 show the recessive phenotype. What is the conclusion of a two-sided binomial test using the normal distribution to approximate the binomial?

A) Fail to reject null hypothesis, we lack evidence that anything other than usual Mendelian segregation is occurring.
B) Fail to reject null hypothesis, we have evidence that something other than usual Mendelian segregation is occurring.
C) Reject null hypothesis, we lack evidence that anything other than usual Mendelian segregation is occurring.
D) Reject null hypothesis, we have evidence that something other than usual Mendelian segregation is occurring.

Question 30

When mating two heterozygotes for alleles in which one is dominant to the other, if the usual Mendelian segregation process is occurring, the ratio of the offspring phenotypes produced should be 3:1. The dominant phenotype would be more common than the recessive one. Imagine a situation in which two heterozygotes are mated and among their 200 offspring, 165 show the dominant phenotype while 35 show the recessive phenotype. What is the P-value of a two-sided binomial test using the normal distribution to approximate the binomial?

A) 0.018
B) 0.036
C) 0.054
D) 0.072

Question 31

When mating two heterozygotes for alleles in which one is dominant to the other, if the usual Mendelian segregation process is occurring, the ratio of the offspring phenotypes produced should be 3:1. The dominant phenotype would be more common than the recessive one. Imagine a situation in which two heterozygotes are mated and among their 200 offspring, 165 show the dominant phenotype while 35 show the recessive phenotype. What is the conclusion of a two-sided binomial test using the normal distribution to approximate the binomial?

A) Fail to reject null hypothesis, we lack evidence that anything other than usual Mendelian segregation is occurring.
B) Fail to reject null hypothesis, we have evidence that something other than usual Mendelian segregation is occurring.
C) Reject null hypothesis, we lack evidence that anything other than usual Mendelian segregation is occurring.
D) Reject null hypothesis, we have evidence that something other than usual Mendelian segregation is occurring.

Question 32

The term for the observation that the medical conditions of patients often improve when they are given inactive chemical substances is called which of the following?

A) Confidence-based improvement
B) Dose response improvement
C) Placebo effect
D) Positive feedback

Question 33

The Declaration of Helsinki states that when testing a new treatment, control patients _______.

A) must be compensated financially to prevent abuse.
B) should receive an inactive yet safe chemical.
C) should receive no chemicals or drugs at all.
D) should receive the best current treatment.

Question 34

To calculate probabilities using the normal probability distribution, we look at the area under the curve between the values we are interested in.

Question 35

The equation for a normal distribution with a mean of 20 and a variance of 30 is
$f(x)=\frac{1}{\sqrt{30 \pi}} e^{\frac{-(x-20)^{2}}{60}}$

Question 36

The equation for a normal distribution with a mean of 30 and a variance of 20 is
$f(x)=\frac{1}{\sqrt{40 \pi}} e^{\frac{-(x-30)^{2}}{40}}$

Question 37

For a normal distribution, the IQR is larger than two standard deviations.

Question 38

The standard normal distribution has a mean of zero and variance of one.

Question 39

The values in normal distribution tables correspond to the probability that the value indicated will be chosen in a random sample.

Question 40

The standard normal deviate describes how many standard deviations away from the mean that a value is.

Question 41

The central limit theorem states that the means of a large number of samples from a population is approximately normally distributed.

Question 42

Using the normal approximation to the binomial distribution is most accurate when the proportion is closest to 0.5.

Question 43

Using the normal distribution instead of the exact binomial distribution is slightly more accurate when sample sizes get very large.

Question 44

The placebo effect appears equally effective at improving a wide variety of conditions, including pain, cholesterol levels, and blood pressure.

Question 45

For a population of 180 values that exhibit a normal distribution in which the mean is 20 and the variance is 6, answer the following questions. What are the mean, median, mode, standard deviation, coefficient of variation, and IQR? How many values are between 18 and 22 and how many are larger than 25?

Question 46

For a population of 240 values that exhibit a normal distribution in which the mean is 75 and the variance is 11, answer the following questions. What are the mean, median, mode, standard deviation, coefficient of variation, and IQR? How many values are between 76 and 80 and how many are larger than 90?

Question 47

Carlos says that he thinks that while humans are more right-handed than left-handed, the same probably isn't true for gorillas. Luis disagrees and says they probably are asymmetric too, but maybe they're more left-handed. Imagine they collect handedness data for 60 gorillas based on videos of the gorillas eating and 37 of the gorillas appear to be right-handed. Use the normal distribution to approximate the binomial distribution to answer this question. Does the data support Carlos or Luis? As part of your answer, present the test statistic and the P-value it corresponds to.

Question 48

Carlos says that he thinks that while humans are more right-handed than left-handed, the same probably isn't true for gorillas. Luis disagrees and says they probably are asymmetric too, but maybe they're more left-handed. Imagine they collect handedness data for 60 gorillas based on videos of the gorillas eating and 39 of the gorillas appear to be right-handed. Use the normal distribution to approximate the binomial distribution to answer this question. Does the data support Carlos or Luis? As part of your answer, present the test statistic and the P-value it corresponds to.

Question 49

Explain the primary benefit of using the normal distribution to model the binomial distribution when calculating P-values.

Question 50

Explain why we should use a correction for continuity when using the normal distribution to model the binomial distribution.

Question 51

Imagine that your friend comes down with a cold and after being sick for a few days takes a supplement called "essence of wormroot" and gets better soon afterward. She then credits the supplement for her recovery. Based on what you know about the likely mechanism of how placebos work, what is the likely explanation for her recovery?