Q: The heights of adult women are normally distributed with mean $$\mu = 160 cm $$and standard deviation $$\sigma = 7 cm. $$Find the height $$x$$ such that 5% of women are shorter than $$x.$$

$$x = 160 + (-1.64) \times 7 = 148.52 cm$$

Question 1

Which of the following statements about the standard normal distribution is TRUE?

Accepted Answer

The mean is 0 and the standard deviation is 1.

Question 2

Suppose the area to the left of $$z = 1.21$$ under the standard normal curve is needed. What is the correct probability to look up in the z-table?

Accepted Answer

$$P(Z \u003c 1.21)$$

Question 3

If $$P(Z \u003e 2.27) is $$required, and the z-table gives cumulative areas from the left, how do you find this probability?

Accepted Answer

Look up $$z = 2.27$$ and subtract the value from 1.

Question 4

Given $$P(Z \u003c z) = 0.6331$$, what is the corresponding $$z-$$score?

Accepted Answer

$$z = 0.34$$

Question 5

A local call center tracks the response time for agents to answer calls, which is uniformly distributed between 2 and 12 seconds. What is the probability that a call is answered in 5 to 9 seconds?

Accepted Answer

$$P(5 \u003c X \u003c 9) = \frac{9-5}{12-2} = 0.4$$

Question 6

The heights of adult women are normally distributed with mean $$\mu = 160 cm $$and standard deviation $$\sigma = 7 cm. $$Find the height $$x$$ such that 5% of women are shorter than $$x.$$

Accepted Answer

$$x = 160 + (-1.64) 	imes 7 = 148.52 cm$$

Question 7

A commuter train arrives every 30 minutes. If a passenger arrives at a random time, what is the probability they wait less than 10 minutes?

Accepted Answer

$$P(X \u003c 10) = \frac{10}{30} = 0.333$$

Question 8

If the mean commute time is 20 minutes with a standard deviation of 5 minutes, what is the $$z-$$score for a commute time of 10 minutes?

Accepted Answer

$$z = \frac{10 - 20}{5} = -2$$

Question 9

Which of the following is a property of a valid probability density function (PDF) for a continuous random variable?

Accepted Answer

The total area under the curve is 1.

Question 10

A sample of 40 players has a sampling distribution mean of 26.7 hours. What is the value of $$n in $$this context?

Accepted Answer

$$n = 40$$

Question 11

According to the Central Limit Theorem, as the sample size $$n$$ increases, the sampling distribution of the sample mean:

Accepted Answer

Becomes more normal, regardless of the population distribution.

Question 12

If $$\mu = 3.2$$, $$\sigma = 0.98$$, and $$n = 60$$, what is the probability of getting a sample mean above 3.5?

Accepted Answer

First, calculate $$z = \frac{3.5 - 3.2}{0.98} = 0.31$$, then use the z-table to find $$P(Z \u003e 0.31) = 0.3783$$

Question 13

When using the normal distribution to approximate a binomial probability, what must you check before proceeding?

Accepted Answer

Both $$np \geq 5$$ and $$nq \geq 5$$

Question 14

A previous study found that 80% of people preferred Pepsi. In a sample of 100, what is the probability at least 60 prefer Pepsi (using normal approximation)?

Accepted Answer

Apply continuity correction: $$P(X \geq 59.5)$$, then find $$z = \frac{59.5 - 80}{\sqrt{100 	imes 0.8 	imes 0.2}} = -4.57$$

Question 15

Which formula is used to convert a non-standard normal variable $$X to a $$standard normal variable $$Z$$?

Accepted Answer

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

Question 16

A used car dealer has 76 cars produced in 2023, and 35 million out of 94 million cars were recalled. What is the probability that more than half of these cars were recalled? (Use normal approximation)

Accepted Answer

Find $$p = \frac{35}{94}$$, $$n = 76$$, apply continuity correction, and use $$z = \frac{38.5 - np}{\sqrt{npq}}$$

Normal Probability Distributions and Sampling Distributions: Study Notes

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