Question 1

A city’s average daily temperature in July is normally distributed with a mean of 85°F and a standard deviation of 4°F. What is the probability that a randomly selected day in July has a temperature above 90°F? Use the standard normal table and round your z-score to two decimal places.

Accepted Answer

$$P(x \u003e 90) = 1 - P(z \u003c 1.25) = 1 - 0.8944 = 0.1056$$

Question 2

Given a normal distribution with mean $$\mu = 67$$ and standard deviation $$\sigma = 3.5$$, what is the z-score for $$x = 70$$? Use the z-score formula.

Accepted Answer

$$z = \frac{70 - 67}{3.5} = 0.86$$

Question 3

A hospital administrator wants to know the probability that a randomly selected state has fewer than 300 physicians per 100,000 residents. The number of physicians is normally distributed with mean $$280$$ and standard deviation $$78. $$What is the probability, rounded to four decimal places, that a state has fewer than 300 physicians per 100,000 residents?

Accepted Answer

$$P(x \u003c 300) = P(z \u003c 0.26) = 0.6026$$

Question 4

Suppose the mean number of physicians per 100,000 residents is $$280$$ with a standard deviation of $$78. $$What is the probability that a randomly selected state has between 300 and 350 physicians per 100,000 residents? Use cumulative probabilities $$P(x \u003c 350) = 0.8153$$ and $$P(x \u003c 300) = 0.6012.$$

Accepted Answer

$$P(300 \u003c x \u003c 350) = 0.8153 - 0.6012 = 0.2141$$

Question 5

Which of the following best describes the process for finding the probability that a normally distributed variable $$x$$ exceeds a certain value $$a$$ using the standard normal table?

Accepted Answer

Convert $$a to a z-$$score, look up the cumulative probability for $$z$$, and subtract this value from 1.

Question 6

A researcher uses technology to find the probability that a normally distributed variable $$x is $$less than $$b. $$Which of the following inputs are required for the calculation?

Accepted Answer

The value of $$x = b$$, the mean $$\mu$$, the standard deviation $$\sigma$$, and an indication for cumulative probability.

Question 7

If the cumulative probability for $$z = 1.20 is$$ $$0.8849$$, what is the probability that a standard normal variable $$z is $$greater than $$1.20$$?

Accepted Answer

$$P(z \u003e 1.20) = 1 - 0.8849 = 0.1151$$

Question 8

A company’s product weights are normally distributed with mean $$50$$ grams and standard deviation $$2$$ grams. What is the probability that a randomly selected product weighs less than $$53$$ grams?

Accepted Answer

$$z = \frac{53 - 50}{2} = 1.5$$, so $$P(x \u003c 53) = P(z \u003c 1.5) = 0.9332$$

Question 9

Why is it necessary to convert a value $$x$$ from a normal distribution to a z-score before using the standard normal table?

Accepted Answer

Because the standard normal table only provides probabilities for the standard normal distribution ($$\mu = 0$$, $$\sigma = 1).$$

Question 10

A state has 350 physicians per 100,000 residents. Given a normal distribution with mean $$280$$ and standard deviation $$78$$, what is the z-score for this state?

Accepted Answer

$$z = \frac{350 - 280}{78} = 0.897$$

Question 11

If you want to find the probability that a normally distributed variable $$x$$ falls between two values $$a$$ and $$b$$ ($$a \u003c b$$), which of the following is the correct approach?

Accepted Answer

Find $$P(x \u003c b)$$ and $$P(x \u003c a)$$, then subtract: $$P(a \u003c x \u003c b) = P(x \u003c b) - P(x \u003c a).$$

Question 12

A speed limit is 70 mph. Vehicle speeds are normally distributed with mean $$67$$ mph and standard deviation $$3.5$$ mph. What is the probability that a randomly selected vehicle is violating the speed limit? (Use $$z = 0.86$$ and $$P(z \u003c 0.86) = 0.8051.)$$

Accepted Answer

$$P(x \u003e 70) = 1 - 0.8051 = 0.1949$$

Question 13

Which of the following statements about cumulative probabilities in the standard normal table is correct?

Accepted Answer

They represent the probability that $$z is $$less than or equal to a given value.

Question 14

A university uses technology to compute the probability that a student’s test score (normally distributed with mean $$75$$ and standard deviation $$8) is $$less than $$90. $$Which Excel function would be appropriate?

Accepted Answer

NORM.DIST(90, 75, 8, TRUE)

Question 15

If you use technology and do not round the z-score, how does the probability result compare to using a rounded z-score from the standard normal table?

Accepted Answer

The probability is slightly more accurate and may differ by a small amount.

Question 16

A normal distribution has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the probability that a randomly selected value is less than $$130$$? (Use $$z = 2.00$$ and $$P(z \u003c 2.00) = 0.9772.)$$

Accepted Answer

$$P(x \u003c 130) = 0.9772$$

Finding Probabilities for Normally Distributed Variables Using Tables and Technology

스터디 가이드 - 연습 문제

학습 가이드 - 플래시카드