Q: Given a normal distribution with $$\mu = 400.3$$ feet and $$\sigma = 25.6$$ feet, what is the 90th percentile value for home run distance?

$$x = 400.3 + (1.28 \times 25.6) = 432.1$$ feet

Q: If $$X is a $$binomial random variable with $$n = 200$$ and $$p = 0.085$$, what is the mean and standard deviation for the normal approximation?

$$\mu = np = 17$$, $$\sigma = \sqrt{np(1-p)} = \sqrt{200 \times 0.085 \times 0.915} \approx 3.96$$

Question 1

Suppose the time (in minutes) it takes for a package to be delivered is uniformly distributed between 0 and 60 minutes. What is the probability that the package arrives between 15 and 30 minutes after 10 am?

Accepted Answer

$$P(15 \u003c X \u003c 30) = \frac{30-15}{60-0} = \frac{15}{60} = 0.25$$

Question 2

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

Accepted Answer

The area under the curve is always less than 1.

Question 3

If a normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15$$, what is the probability that a randomly selected value falls between $$\mu - \sigma$$ and $$\mu + \sigma$$?

Accepted Answer

Approximately 68%

Question 4

A factory produces ball bearings with diameters that are normally distributed with mean $$\mu = 1 cm $$and standard deviation $$\sigma = 0.002 cm. $$What is the $$z-$$score for a ball bearing with diameter $$0.995 cm$$?

Accepted Answer

$$z = \frac{0.995 - 1}{0.002} = -2.5$$

Question 5

Which statement best describes the role of area under the normal curve?

Accepted Answer

It represents the probability of observing a value within a given interval.

Question 6

Given a normal distribution with $$\mu = 400.3$$ feet and $$\sigma = 25.6$$ feet, what is the 90th percentile value for home run distance?

Accepted Answer

$$x = 400.3 + (1.28 	imes 25.6) = 432.1$$ feet

Question 7

Which of the following best describes the Empirical Rule for a normal distribution?

Accepted Answer

About 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3 standard deviations of the mean.

Question 8

A sample of 17 workers' FEV/VC ratios is plotted on a normal probability plot. The linear correlation coefficient between observed values and expected Z-scores is 0.950. According to Table VI, is it reasonable to conclude the data are from a normal population?

Accepted Answer

Yes, because $$0.950 \u003e 0.946 ($$the critical value for $$n=17)$$

Question 9

Which of the following is a criterion for using the normal approximation to the binomial distribution?

Accepted Answer

$$np(1-p) \geq 10$$

Question 10

If $$X is a $$binomial random variable with $$n = 200$$ and $$p = 0.085$$, what is the mean and standard deviation for the normal approximation?

Accepted Answer

$$\mu = np = 17$$, $$\sigma = \sqrt{np(1-p)} = \sqrt{200 	imes 0.085 	imes 0.915} \approx 3.96$$

Question 11

A normal probability plot is approximately linear for a data set. What does this indicate about the data?

Accepted Answer

The data are likely from a population that is normally distributed.

Question 12

If the area under a normal curve to the right of $$x = 450$$ feet is $$0.026$$, what does this mean in the context of home run distances?

Accepted Answer

2.6% of home runs traveled more than 450 feet.

Question 13

Given a normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$, what is the $$z-$$score for a value $$x$$?

Accepted Answer

$$z = \frac{x - \mu}{\sigma}$$

Question 14

A normal distribution has mean $$\mu = 50$$ and standard deviation $$\sigma = 5. $$What value of $$x$$ corresponds to the 97.5th percentile?

Accepted Answer

$$x = 50 + (1.96 	imes 5) = 59.8$$

Question 15

Which of the following best describes the inflection points on a normal curve?

Accepted Answer

They occur at $$\mu - \sigma$$ and $$\mu + \sigma.$$

Chapter 7: The Normal Probability Distribution – Study Notes

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