Q: A university only accepts students whose SAT verbal scores are in the top 4%. If SAT scores are normally distributed with mean 500 and standard deviation 100, what is the minimum score required for acceptance? (Use $$z = 1.75$$ for the 96th percentile.)

$$x = 500 + 1.75 \times 100 = 675$$

Question 1

Which of the following is the correct formula for the probability density function of a normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

$$f(x) = \frac{1}{\sqrt{2\pi\$$sigma^2$$}} e$$^{-\frac{(x-\mu)^2}$${2\$$sigma^2$$}}$$

Question 2

A standard normal random variable $$Z$$ has which of the following properties?

Accepted Answer

$$\mu = 0$$ and $$\$$sigma^2$$ = 1$$

Question 3

According to the Empirical Rule, approximately what percentage of data falls within two standard deviations of the mean in a normal distribution?

Accepted Answer

95%

Question 4

If $$X is a $$normal random variable with mean $$\mu$$ and standard deviation $$\sigma$$, what is the $$z-$$score for a value $$x$$?

Accepted Answer

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

Question 5

A light bulb has a lifetime that is normally distributed with mean 1100 hours and standard deviation 100 hours. What is the $$z-$$score for a bulb that lasts 1200 hours?

Accepted Answer

$$z = 1$$

Question 6

Which of the following statements is TRUE about the area under the standard normal curve?

Accepted Answer

The total area under the curve is 1.

Question 7

If the area to the left of $$z is 0.975$$, what is the value of $$z ($$rounded to two decimal places)?

Accepted Answer

$$z = 1.96$$

Question 8

A pizza delivery time is normally distributed with mean 25 minutes and standard deviation 2 minutes. What is the probability that a delivery takes less than 22.7 minutes? (Find the $$z-$$score first.)

Accepted Answer

$$z = \frac{22.7 - 25}{2} = -1.15$$

Question 9

Which of the following best describes the effect of increasing the variance $$\$$sigma^2$$ of a normal distribution while keeping the mean $$\mu$$ constant?

Accepted Answer

The curve becomes wider and shorter.

Question 10

Suppose the area to the left of $$z is 0.01. $$What is the value of $$z ($$rounded to two decimal places)?

Accepted Answer

$$z = -2.33$$

Question 11

If $$P(Z \u003e z) = 0.05$$ for a standard normal variable $$Z$$, what is the value of $$z ($$rounded to two decimal places)?

Accepted Answer

$$z = 1.645$$

Question 12

A university only accepts students whose SAT verbal scores are in the top 4%. If SAT scores are normally distributed with mean 500 and standard deviation 100, what is the minimum score required for acceptance? (Use $$z = 1.75$$ for the 96th percentile.)

Accepted Answer

$$x = 500 + 1.75 	imes 100 = 675$$

Question 13

The cumulative distribution function $$\Phi(z)$$ for the standard normal variable $$Z$$ gives:

Accepted Answer

The probability that $$Z is $$less than or equal to $$z.$$

Question 14

If $$P(-z \u003c Z \u003c z) = 0.99$$ for a standard normal variable $$Z$$, what is the value of $$z ($$rounded to two decimal places)?

Accepted Answer

$$z = 2.58$$

Question 15

A shaft diameter is normally distributed with mean 0.2508 inch and standard deviation 0.0005 inch. What is the $$z-$$score for a shaft measuring 0.2523 inch?

Accepted Answer

$$z = \frac{0.2523 - 0.2508}{0.0005} = 3$$

Question 16

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

Accepted Answer

It is skewed to the right.

Question 17

If $$P(Z \u003c z) = 0.0051$$, what is the value of $$z ($$rounded to two decimal places)?

Accepted Answer

$$z = -2.58$$

Question 18

Which of the following best describes the process of standardizing a normal random variable $$X$$?

Accepted Answer

Subtract the mean and divide by the standard deviation.

Question 19

If $$P(Z \u003c z) = 0.9900$$, what is the value of $$z ($$rounded to two decimal places)?

Accepted Answer

$$z = 2.33$$

Study Notes: The Normal Distribution in Introductory Statistics

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