Question 1

Using the following uniform density curve, answer the question. 
-What is the probability that the random variable has a value greater than 5?

Accepted Answer

The probability that the random variable has a value greater than 5 is 0.375. This is because the random variable has a uniform distribution between 0 and 10, and the probability of any value between 0 and 10 is 1/10. Therefore, the probability of a value greater than 5 is 5/10 = 0.375.

Question 2

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 4%. Find the reading that separates the bottom 4% from the others.

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

Samples of size n = 900 are randomly selected from the population of numbers (0 through 9) produced by a random-number generator, and the proportion of even numbers is found for each sample. What is the distribution of the sample proportions?

Accepted Answer

The distribution of sample proportions will be approximately normal due to the Central Limit Theorem, as the sample size is large (n = 900).

Question 5

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 at least 42 boys

Accepted Answer

The probability of at least 42 boys corresponds to the area to the right of 41.5, because the continuity correction adjusts the discrete value of 42 to 41.5.

Question 6

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
-$\text { With } n = 18 \text { and } p = 0.30 \text {, estimate } \mathrm { P } ( 6 ) \text {. }$

Accepted Answer

The answer of Estimate the indicated probability by using the...

Question 7

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
-A product is manufactured in batches of 120 and the overall rate of defects is 5%. Estimate the probability that a randomly selected batch contains more than 6 defects.

Accepted Answer

The answer of Estimate the indicated probability by using the...

Question 8

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 76.

Accepted Answer

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

Question 9

In a population of 225 women, the heights of the women are normally distributed with a mean of 64.5 inches and a standard deviation of 2.9 inches. If 25 women are selected at random, find the probability that their mean height will exceed 66 inches. Assume that the sampling is done without replacement and use a finite population correction factor with $\mathrm { N } = 225$

Accepted Answer

The probability is calculated using the finite population correction factor and the standard error of the mean. The corrected standard error is smaller due to the finite population, and using the z-score formula, the probability of the sample mean exceeding 66 inches is found to be approximately 0.0031.

Question 10

Assume that z scores are normally distributed with a mean of 0 and a standard deviation of 1. If $\mathrm { P } ( 0 < \mathrm { z } < \mathrm { a } ) = 0.4608 \text {, find } a \text {. }$

Accepted Answer

The answer of Assume that z scores are normally distributed...

Question 11

A baseball player has a batting average of 0.346, so the probability of a hit is 0.346. Assume that his hitting attempts are independent of each other. Assume that the batter gets up to bat 4 times in each game. Estimate the probability that in 50 consecutive games, there are at least 45 games in which the batter gets at least one hit. (Hint: first find the probability that in one game the batter gets at least one hit)

Accepted Answer

The answer of A baseball player has a batting average...

Question 12

Using the following uniform density curve, answer the question. 
-What is the probability that the random variable has a value between 2.3 and 5?

Accepted Answer

The probability that the random variable has a value between 2.3 and 5 is the area under the normal curve between those two values. This area can be found using a standard normal distribution table or a calculator. The area between 2.3 and 5 is 0.3375.

Question 13

If z is a standard normal variable, find the probability.
-The probability that z is less than 1.13

Accepted Answer

The probability that a standard normal variable z is less than 1.13 is approximately 0.8708, based on standard normal distribution tables or a calculator.

Question 14

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 high, but all other thermometers are acceptable, find the temperature that separates the rejected thermometers from the others.

Accepted Answer

The z-score corresponding to the top 9% of a standard normal distribution is approximately 1.34. Therefore, the temperature reading that separates the rejected thermometers from the others is 1.34°C.

Question 15

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.
-$$\mathrm { n } = 20 \text { and } \mathrm { p } = 0.6$$

Accepted Answer

The normal approximation is suitable if both np and n(1-p) are greater than or equal to 5. Here, np = 20 * 0.6 = 12 and n(1-p) = 20 * 0.4 = 8, both of which are greater than 5.

Question 16

The diameters of bolts produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. What percentage of bolts will have a diameter greater than 0.32 inches?

Accepted Answer

The z-score for a diameter of 0.32 inches is (0.32 - 0.30) / 0.01 = 2. A z-score of 2 corresponds to the 97.72 percentile, meaning 97.72% of bolts have a diameter less than 0.32 inches. Therefore, 100% - 97.72% = 2.28% of bolts have a diameter greater than 0.32 inches.

Question 17

Scores on a test are normally distributed with a mean of 68.9 and a standard deviation of 11.6. Find P81, which separates the bottom 81% from the top 19%.

Accepted Answer

P81 corresponds to the 81st percentile of a normal distribution. Using the z-score table, the z-score for 0.81 is approximately 0.88. Using the formula X = μ + zσ, where μ is the mean and σ is the standard deviation, we calculate X = 68.9 + 0.88(11.6) = 79.1.

Question 18

If a continuous uniform distribution has parameters of $\mu = 0$ and $\sigma = 1$, then the minimum is $- \sqrt { 3 }$ and the maximum is $\sqrt { 3 }$. For this distribution, find $\mathrm { P } ( - 1 < x < 1.5 )$. Round your answer to three decimal places.

Accepted Answer

The probability density function of a continuous uniform distribution is given by:$$f(x) = \frac{1}{b-a}, \quad a \le x \le b$$where $a$ and $b$ are the minimum and maximum values, respectively.In this case, $\mu = 0$ and $\sigma = 1$, so the minimum is $- \sqrt { 3 }$ and the maximum is $\sqrt { 3 }$. Therefore, the probability density function is:$$f(x) = \frac{1}{2\sqrt{3}}, \quad - \sqrt{3} \le x \le \sqrt{3}$$The probability of $x$ being between -1 and 1.5 is:$$P(-1 < x < 1.5) = \int_{-1}^{1.5} \frac{1}{2\sqrt{3}} dx = \frac{1}{2\sqrt{3}} \left[ x \right]_{-1}^{1.5} = \frac{1}{2\sqrt{3}} (1.5 - (-1)) = \frac{1}{2\sqrt{3}} (2.5) = 0.722$$Therefore, the correct answer is A.

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 that the number of correct answers is between 30 and 59 inclusive

Accepted Answer

The continuity correction involves adjusting the discrete interval [30, 59] to the continuous interval [29.5, 59.5] to better approximate the normal distribution.

Question 20

Find the indicated value.
-$\mathrm { z } _ { 0.005 }$

Accepted Answer

The answer of Find the indicated value.
-\(\mathrm { z }...

Quiz 6: Normal Probability Distributions

Using the following uniform density curve, answer the question. -What is the probability that the random variable has a value greater than 5?

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

Samples of size n = 900 are randomly selected from the population of numbers (0 through 9) produced by a random-number generator, and the proportion of even numbers is found for each sample. What is the distribution of the sample proportions?

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 at least 42 boys

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. - $\text { With } n = 18 \text { and } p = 0.30 \text {, estimate } \mathrm { P } ( 6 ) \text {. }$

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. -A product is manufactured in batches of 120 and the overall rate of defects is 5%. Estimate the probability that a randomly selected batch contains more than 6 defects.

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 76.

Assume that z scores are normally distributed with a mean of 0 and a standard deviation of 1. If $\mathrm { P } ( 0 < \mathrm { z } < \mathrm { a } ) = 0.4608 \text {, find } a \text {. }$

Using the following uniform density curve, answer the question. -What is the probability that the random variable has a value between 2.3 and 5?

If z is a standard normal variable, find the probability. -The probability that z is less than 1.13

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. - $\mathrm { n } = 20 \text { and } \mathrm { p } = 0.6$

The diameters of bolts produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. What percentage of bolts will have a diameter greater than 0.32 inches?

Scores on a test are normally distributed with a mean of 68.9 and a standard deviation of 11.6. Find P81, which separates the bottom 81% from the top 19%.

If a continuous uniform distribution has parameters of $\mu = 0$ and $\sigma = 1$ , then the minimum is $- \sqrt { 3 }$ and the maximum is $\sqrt { 3 }$ . For this distribution, find $\mathrm { P } ( - 1 < x < 1.5 )$ . Round your answer to three decimal places.

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 that the number of correct answers is between 30 and 59 inclusive

Find the indicated value. - $\mathrm { z } _ { 0.005 }$

Quiz 6: Normal Probability Distributions

Using the following uniform density curve, answer the question. -What is the probability that the random variable has a value greater than 5?

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

Samples of size n = 900 are randomly selected from the population of numbers (0 through 9) produced by a random-number generator, and the proportion of even numbers is found for each sample. What is the distribution of the sample proportions?

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 at least 42 boys

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. - With n=18 and p=0.30, estimate P(6) . \text { With } n = 18 \text { and } p = 0.30 \text {, estimate } \mathrm { P } ( 6 ) \text {. } With n=18 and p=0.30, estimate P(6) .

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. -A product is manufactured in batches of 120 and the overall rate of defects is 5%. Estimate the probability that a randomly selected batch contains more than 6 defects.

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 76.

Assume that z scores are normally distributed with a mean of 0 and a standard deviation of 1. If P(0<z<a) =0.4608, find a. \mathrm { P } ( 0 < \mathrm { z } < \mathrm { a } ) = 0.4608 \text {, find } a \text {. }P(0<z<a) =0.4608, find a.

Using the following uniform density curve, answer the question. -What is the probability that the random variable has a value between 2.3 and 5?

If z is a standard normal variable, find the probability. -The probability that z is less than 1.13

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. - n=20 and p=0.6\mathrm { n } = 20 \text { and } \mathrm { p } = 0.6n=20 and p=0.6

The diameters of bolts produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. What percentage of bolts will have a diameter greater than 0.32 inches?

Scores on a test are normally distributed with a mean of 68.9 and a standard deviation of 11.6. Find P81, which separates the bottom 81% from the top 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 that the number of correct answers is between 30 and 59 inclusive

Find the indicated value. - z0.005\mathrm { z } _ { 0.005 }z0.005​

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. - $\text { With } n = 18 \text { and } p = 0.30 \text {, estimate } \mathrm { P } ( 6 ) \text {. }$

Assume that z scores are normally distributed with a mean of 0 and a standard deviation of 1. If $\mathrm { P } ( 0 < \mathrm { z } < \mathrm { a } ) = 0.4608 \text {, find } a \text {. }$

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. - $\mathrm { n } = 20 \text { and } \mathrm { p } = 0.6$

Find the indicated value. - $\mathrm { z } _ { 0.005 }$