Question 1

For the binomial distribution with the given values for n and p, state whether or not it is suitable to use the normal distribution as an approximation.
-$\mathrm { n } = 17 \text { and } \mathrm { p } = 0.5$
A) Normal approximation is suitable.
B) Normal approximation is not suitable.

Accepted Answer

The normal approximation is suitable when both np and n(1-p) are greater than or equal to 5. Here, np = 17 * 0.5 = 8.5 and n(1-p) = 17 * 0.5 = 8.5, both of which are greater than 5.

Question 2

Assume that X has a normal distribution, and find the indicated probability.
-The mean is $\mu = 15.2$ and the standard deviation is $\sigma = 0.9$. Find the probability that $X$ is greater than 17 .
A) 0.9821
B) 0.9772
C) 0.0228
D) 0.9713

Accepted Answer

The answer of Assume that X has a normal distribution,...

Question 3

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
-Estimate the probability of getting exactly 43 boys in 90 births.
A) 0.1628
B) 0.0764
C) 0.0159
D) 0.0729

Accepted Answer

To use the normal approximation to the binomial distribution, we first calculate the mean (μ) and standard deviation (σ) of the binomial distribution. Here, n = 90 and p = 0.5 (assuming equal probability for boys and girls). Thus, μ = np = 45 and σ = sqrt(np(1-p)) = sqrt(22.5) ≈ 4.74. We then use the continuity correction for P(X = 43) by finding P(42.5 < X < 43.5). Converting to the standard normal distribution, we find the z-scores and calculate the probability, which is approximately 0.0764.

Question 4

Round to the nearest tenth unless indicated otherwise
-The amount of rainfall in January in a certain city is normally distributed with a mean of 4.9 inches and a standard deviation of 0.3 inches. Find the value of the quartile $Q _ { 1 }$ .
A) 4.8
B) 5.1
C) 1.2
D) 4.7

Accepted Answer

The first quartile $Q_1$ is the 25th percentile of a normal distribution. Using the Z-score table, the Z-score for the 25th percentile is approximately -0.675. Using the formula $Q_1 = \mu + Z \cdot \sigma$, we get $Q_1 = 4.9 + (-0.675) \cdot 0.3 = 4.7$.

Question 5

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.
-If 9% of the thermometers are rejected because they have readings that are too low, but all other thermometers are acceptable, find the temperature that separates the rejected thermometers from the others. 
A) $- 1.26 ^ { \circ }$
B) $- 1.39 ^ { \circ }$
C) $- 1.45 ^ { \circ }$
D) $- 1.34 ^ { \circ }$

Accepted Answer

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

Question 6

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.
-If 7% of the thermometers are rejected because they have readings that are too low, but all other thermometers are acceptable, find the temperature that separates the rejected thermometers from the others. 
A) $- 1.53 ^ { \circ }$
B) $- 1.26 ^ { \circ }$
C) $- 1.48 ^ { \circ }$
D) $- 1.39 ^ { \circ }$

Accepted Answer

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

Question 7

If z is a standard normal variable, find the probability.
-The probability that z lies between $- 0.55 \text { and } 0.55$
A) 0.9000
B) 0.4176
C) -0.9000
D) -0.4176

Accepted Answer

The probability that a standard normal variable z lies between -0.55 and 0.55 is found using the standard normal distribution table. The probability for z < 0.55 is approximately 0.7088, and for z < -0.55 is approximately 0.2912. The difference, 0.7088 - 0.2912, gives the probability of z being between -0.55 and 0.55, which is 0.4176.

Question 8

Scores on a test are normally distributed with a mean of 68.2 and a standard deviation of 10.4. Estimate the probability that among 75 randomly selected students, at least 20 of them score greater than 78.
A) 0.1736
B) 0.0113
C) 0.0166
D) 0.0239

Accepted Answer

The answer of Scores on a test are normally distributed...

Question 9

The weights of certain machine components are normally distributed with a mean of 8.6 g and a standard deviation of 0.06 g. Find the two weights that separate the top 3% and the bottom 3%. these weights could serve as limits used to identify which components should be rejected. Round to the nearest hundredth of a gram.
A) 8.49 g and 8.71 g
B) 8.59 g and 8.61 g
C) 8.57 g and 8.63 g
D) 8.47 g and 8.76 g

Accepted Answer

The z-scores for the top 3% and the bottom 3% are -1.88 and 1.88, respectively. The corresponding weights are 8.49 g and 8.71 g.

Question 10

Assume that X has a normal distribution, and find the indicated probability.
-The mean is $\mu = 15.2$ and the standard deviation is $\sigma = 0.9$. Find the probability that $X$ is greater than $15.2$.
A) 0.9998
B) 1.0000
C) 0.5000
D) 0.0003

Accepted Answer

Since the mean of a normal distribution is the point of symmetry, the probability that $X$ is greater than the mean is 0.5.

Question 11

For women aged $18 - 24$, systolic blood pressures (in $\mathrm { mm } \mathrm { Hg }$ ) are normally distributed with a mean of $114.8$ and a standard deviation of 13.1. If 36 women are selected at random from a population of 300 women aged 18-24, find the probability that their mean systolic blood pressure will be less than $110 \mathrm {~mm} \mathrm { Hg }$. Assume that the sampling is done without replacement and use a finite population correction factor with $\mathrm { N } = 300$.
A) 0.0146
B) 0.0096
C) 0.3557
D) 0.3483

Accepted Answer

To find the probability that the sample mean is less than 110 mm Hg, we first calculate the standard error of the mean using the finite population correction factor. The standard error is given by $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \times \sqrt{\frac{N-n}{N-1}}$, where $\sigma = 13.1$, $n = 36$, and $N = 300$. This gives $\sigma_{\bar{x}} \approx 2.06$. The z-score for 110 is $\frac{110 - 114.8}{2.06} \approx -2.33$. Using the standard normal distribution table, the probability corresponding to a z-score of -2.33 is approximately 0.0096.

Question 12

Using the following uniform density curve, answer the question. 
-What is the probability that the random variable has a value between 0.6 and 0.8?
A) 0.1
B) 0.275
C) 0.15
D) 0.025

Accepted Answer

The answer of Using the following uniform density curve, answer...

Question 13

For the binomial distribution with the given values for n and p, state whether or not it is suitable to use the normal
distribution as an approximation.
-$\mathrm { n } = 19 \text { and } \mathrm { p } = 0.8$
A) Normal approximation is not suitable.
B) Normal approximation is suitable.

Accepted Answer

The answer of For the binomial distribution with the given...

Question 14

Use the normal distribution to approximate the desired probability.
-Find the probability that in 200 tosses of a fair die, we will obtain at exactly 30 fives.
A) 0.1871
B) 0.0429
C) 0.0871
D) 0.0619

Accepted Answer

To approximate the probability using a normal distribution, we use the binomial distribution parameters: n = 200, p = 1/6. The mean (μ) is np = 200*(1/6) = 33.33 and the standard deviation (σ) is sqrt(np(1-p)) = sqrt(200*(1/6)*(5/6)) ≈ 5.27. Using the continuity correction, we find P(29.5 < X < 30.5) and convert to the standard normal distribution to find the probability, which is approximately 0.0619.

Question 15

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.
-Between 7.5 pounds and 9 pounds 
A) $\frac { 3 } { 4 }$

B) $\frac { 1 } { 2 }$

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

D) $\frac { 1 } { 4 }$

Accepted Answer

The range of the uniform distribution is 6 to 12 pounds, so the total range is 6 pounds. The range between 7.5 and 9 pounds is 1.5 pounds. The probability is the length of the interval (1.5) divided by the total range (6), which is $\frac{1.5}{6} = \frac{1}{4}$.

Question 16

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 exceeds 8.1 hours.
A) 0.8531
B) 0.9146
C) 0.8457
D) 0.7285

Accepted Answer

The probability that the mean rebuild time exceeds 8.1 hours is 0.8531. This is because the z-score for a mean rebuild time of 8.1 hours is (8.1 - 8.4) / 1.8 = -0.17. The probability that a z-score is less than -0.17 is 0.8531.

Question 17

Assume that women have heights that are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. Find the value of the quartile $\mathrm { Q } _ { 3 }$ .
A) 66.1 inches
B) 65.3 inches
C) 67.8 inches
D) 64.3 inches

Accepted Answer

The answer of Assume that women have heights that are...

Question 18

If z is a standard normal variable, find the probability.
-The probability that z lies between $- 2.41 \text { and } 0$
A) 0.0948
B) 0.5080
C) 0.4910
D) 0.4920

Accepted Answer

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

Question 19

Use the normal distribution to approximate the desired probability.
-Find the probability that in 200 tosses of a fair die, we will obtain at least 30 fives.
A) 0.5871
B) 0.6229
C) 0.8871
D) 0.7673

Accepted Answer

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

Question 20

Match between columns Premises: If selecting samples of si n=10 from a population with a known mean and standard deviation,
what requirement, if any, must be satisfied in order to assume that the distribution of the sample
means is a normal distribution?

Accepted Answer

The Answer of None; the distribution of sample means will is...

Quiz 6: Normal Probability Distributions

For the binomial distribution with the given values for n and p, state whether or not it is suitable to use the normal distribution as an approximation. - $\mathrm { n } = 17 \text { and } \mathrm { p } = 0.5$

Assume that X has a normal distribution, and find the indicated probability. -The mean is $\mu = 15.2$ and the standard deviation is $\sigma = 0.9$ . Find the probability that $X$ is greater than 17 .

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. -Estimate the probability of getting exactly 43 boys in 90 births.

Round to the nearest tenth unless indicated otherwise -The amount of rainfall in January in a certain city is normally distributed with a mean of 4.9 inches and a standard deviation of 0.3 inches. Find the value of the quartile $Q _ { 1 }$ .

If z is a standard normal variable, find the probability. -The probability that z lies between $- 0.55 \text { and } 0.55$

Scores on a test are normally distributed with a mean of 68.2 and a standard deviation of 10.4. Estimate the probability that among 75 randomly selected students, at least 20 of them score greater than 78.

Assume that X has a normal distribution, and find the indicated probability. -The mean is $\mu = 15.2$ and the standard deviation is $\sigma = 0.9$ . Find the probability that $X$ is greater than $15.2$ .

Using the following uniform density curve, answer the question. -What is the probability that the random variable has a value between 0.6 and 0.8?

For the binomial distribution with the given values for n and p, state whether or not it is suitable to use the normal distribution as an approximation. - $\mathrm { n } = 19 \text { and } \mathrm { p } = 0.8$

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

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. -Between 7.5 pounds and 9 pounds

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 exceeds 8.1 hours.

Assume that women have heights that are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches. Find the value of the quartile $\mathrm { Q } _ { 3 }$ .

If z is a standard normal variable, find the probability. -The probability that z lies between $- 2.41 \text { and } 0$

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

Provide an appropriate response.

Quiz 6: Normal Probability Distributions

For the binomial distribution with the given values for n and p, state whether or not it is suitable to use the normal distribution as an approximation. - n=17 and p=0.5\mathrm { n } = 17 \text { and } \mathrm { p } = 0.5n=17 and p=0.5

Assume that X has a normal distribution, and find the indicated probability. -The mean is μ=15.2\mu = 15.2μ=15.2 and the standard deviation is σ=0.9\sigma = 0.9σ=0.9 . Find the probability that XXX is greater than 17 .