Q: A ball bearing diameter is normally distributed with $$\mu = 1.0 cm $$and $$\sigma = 0.02 cm. $$What diameter corresponds to the 97.5th percentile?

$$1.0 + 1.96 \times 0.02 = 1.0392 cm$$

Q: A normal distribution has mean $$\mu = 50$$ and standard deviation $$\sigma = 5. $$What is the value of $$x$$ that corresponds to $$z = 1.5$$?

$$x = 50 + 1.5 \times 5 = 57.5$$

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

$$np \geq 10$$ \, \text{and} \, n(1-p) \geq 10$

Q: A home run distance is normally distributed with mean $$400 ft $$and standard deviation $$25 ft. $$What is the percentile rank of a home run that traveled $$425 ft$$?

$$z = \frac{425 - 400}{25} = 1$$, \text{so percentile} \approx 84\%$

Question 1

Suppose a uniform probability distribution is used to model the arrival time of a UPS package between 10:00 am and 11:00 am. What is the probability that the package arrives between 10:15 am and 10:45 am?

Accepted Answer

$$P = \frac{30}{60} = 0.5$$

Question 2

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

Accepted Answer

The curve is skewed to the right.

Question 3

A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What proportion of data falls between $$\mu - 2\sigma$$ and $$\mu + 2\sigma$$?

Accepted Answer

$$\approx 95\%$$

Question 4

If the mean of a normal distribution is increased while keeping the standard deviation constant, what happens to the graph of the normal curve?

Accepted Answer

The curve shifts to the right.

Question 5

A factory produces ball bearings with diameters that are normally distributed with mean $$\mu = 1.0 cm $$and standard deviation $$\sigma = 0.02 cm. $$What is the probability that a randomly selected ball bearing has a diameter between $$0.98 cm $$and $$1.02 cm$$?

Accepted Answer

$$\approx 95\%$$

Question 6

Which formula is used to standardize a value $$x$$ from a normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

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

Question 7

A home run distance in baseball is normally distributed with mean $$400 ft $$and standard deviation $$25 ft. $$What is the $$z-$$score for a home run of $$450 ft$$?

Accepted Answer

$$z = \frac{450 - 400}{25} = 2$$

Question 8

Which statement best describes the area under the normal curve between $$\mu - \sigma$$ and $$\mu + \sigma$$?

Accepted Answer

It represents approximately $$68\% of $$the data.

Question 9

A normal probability plot is used to assess normality. If the points on the plot lie close to a straight line, what does this indicate?

Accepted Answer

The data are approximately normally distributed.

Question 10

For a binomial distribution with $$n = 100$$ and $$p = 0.5$$, can the normal approximation be used?

Accepted Answer

Yes, because $$np = 50$$ and $$n(1-p) = 50$$, both greater than 10.

Question 11

A sample of 200 American households is surveyed, and 8.5% are expected to buy their pet dog a gift for Valentine's Day. What is the expected number of households buying a gift?

Accepted Answer

$$200 	imes 0.085 = 17$$

Question 12

If a normal distribution has mean $$\mu$$ and standard deviation $$\sigma$$, what is the probability that a randomly selected value is greater than $$\mu + 3\sigma$$?

Accepted Answer

$$\approx 0.15\%$$

Question 13

A ball bearing diameter is normally distributed with $$\mu = 1.0 cm $$and $$\sigma = 0.02 cm. $$What diameter corresponds to the 97.5th percentile?

Accepted Answer

$$1.0 + 1.96 	imes 0.02 = 1.0392 cm$$

Question 14

Which of the following is the correct formula for the mean of a binomial distribution?

Accepted Answer

$$\mu = np$$

Question 15

A normal distribution has mean $$\mu = 50$$ and standard deviation $$\sigma = 5. $$What is the value of $$x$$ that corresponds to $$z = 1.5$$?

Accepted Answer

$$x = 50 + 1.5 	imes 5 = 57.5$$

Question 16

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

Accepted Answer

$$np \geq 10$$ \, 	ext{and} \, n(1-p) \geq 10$

Question 17

If a normal probability plot of sample data shows a pronounced curve rather than a straight line, what does this suggest?

Accepted Answer

The data are not normally distributed.

Question 18

A home run distance is normally distributed with mean $$400 ft $$and standard deviation $$25 ft. $$What is the percentile rank of a home run that traveled $$425 ft$$?

Accepted Answer

$$z = \frac{425 - 400}{25} = 1$$, 	ext{so percentile} \approx 84\%$

Question 19

Which of the following best describes the shape of a normal distribution?

Accepted Answer

Bell-shaped and symmetric

Chapter 7: The Normal Probability Distribution – Properties, Applications, and Approximations

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