Q: A normal distribution has $$\mu = 80$$ and $$\sigma = 10. $$What value of $$X$$ corresponds to the 97.5th percentile?

$$X = 80 + 1.96 \times 10 = 99.6$$

Question 1

Which of the following is NOT a property of a continuous probability distribution's probability density function (pdf)?

Accepted Answer

The probability of any single exact value is always greater than zero.

Question 2

Suppose the waiting time for a bus is uniformly distributed between 0 and 30 minutes. What is the probability that a person waits more than 20 minutes?

Accepted Answer

$$P(X \u003e 20) = \frac{30-20}{30} = \frac{10}{30} = 0.333$$

Question 3

A normal distribution has mean $$\mu = 50$$ and standard deviation $$\sigma = 5. $$What is the probability that a randomly selected value is between 45 and 55?

Accepted Answer

Approximately 68%

Question 4

Which of the following statements about the normal distribution is FALSE?

Accepted Answer

Changing the mean changes the spread of the curve.

Question 5

Given a normal distribution with $$\mu = 100$$ and $$\sigma = 15$$, what is the $$z-$$score for $$X = 130$$?

Accepted Answer

$$z = \frac{130 - 100}{15} = 2$$

Question 6

The Empirical Rule states that approximately what percentage of data falls within two standard deviations of the mean in a normal distribution?

Accepted Answer

95%

Question 7

If the area under the normal curve to the left of $$x = 2100$$ pounds is 0.3085 for giraffe weights, what does this mean?

Accepted Answer

About 30.85% of giraffes weigh less than 2100 pounds.

Question 8

A manufacturer wants to accept the middle 90% of steel rods produced, which are normally distributed with $$\mu = 100 cm $$and $$\sigma = 0.45 cm. $$What $$z-$$scores correspond to the endpoints of this interval?

Accepted Answer

$$z = \pm 1.645$$

Question 9

Which of the following best describes the purpose of a normal probability plot?

Accepted Answer

To assess whether sample data could come from a population that is normally distributed.

Question 10

If a histogram of sample data is symmetric and bell-shaped, what can you conclude about the underlying distribution?

Accepted Answer

It is likely to be approximately normal.

Question 11

Suppose the number of chocolate chips in a bag is normally distributed with $$\mu = 1262$$ and $$\sigma = 118. $$What is the $$z-$$score for a bag with 1000 chips?

Accepted Answer

$$z = \frac{1000 - 1262}{118} \approx -2.22$$

Question 12

Which of the following is true for any continuous random variable $$X$$?

Accepted Answer

$$P(X = a) = 0$$ for any specific value $$a$$

Question 13

If the area under the standard normal curve to the left of $$z is 0.06$$, what is the approximate value of $$z$$?

Accepted Answer

$$z \approx -1.55$$

Question 14

A normal distribution has $$\mu = 80$$ and $$\sigma = 10. $$What value of $$X$$ corresponds to the 97.5th percentile?

Accepted Answer

$$X = 80 + 1.96 	imes 10 = 99.6$$

Question 15

Which of the following is a correct interpretation of the area under the normal curve between $$z = -1$$ and $$z = 1$$?

Accepted Answer

It represents the probability that a standard normal variable falls within one standard deviation of the mean.

Question 16

If a normal probability plot of sample data is approximately linear and the correlation coefficient is greater than the critical value, what can you conclude?

Accepted Answer

It is reasonable to conclude the data could come from a normal population.

Continuous Probability Distributions and the Normal Distribution

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