Q: A GRE score is normally distributed with mean $$\mu = 1049$$ and standard deviation $$\sigma = 189. $$What score corresponds to the 85th percentile?

$$x = 1049 + 1.04 \times 189 = 1246$$

Question 1

Which of the following statements about the uniform probability distribution shown in the first image is correct?

Accepted Answer

The probability density function is $$f(x) = \frac{1}{60}$$ for $$0 \leq x \leq 60.$$

Question 2

Given a uniform density function $$f(x) = \frac{1}{60}$$ for $$0 \leq x \leq 60$$, what is the probability that the random variable $$X$$ falls between 15 and 30?

Accepted Answer

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

Question 3

Which property is NOT true for a normal density curve?

Accepted Answer

The curve has multiple peaks.

Question 4

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

Accepted Answer

95%

Question 5

If the mean $$\mu = 100$$ and standard deviation $$\sigma = 15$$, what is the $$z-$$score for an individual with $$x = 120$$?

Accepted Answer

$$z = \frac{120 - 100}{15} = 1.33$$

Question 6

What does the area under the normal curve to the left of $$x$$ represent?

Accepted Answer

The proportion of the population with values less than $$x.$$

Question 7

If the area under the standard normal curve to the left of $$z = 1.33 is$$ $$0.9082$$, what is the area to the right of $$z = 1.33$$?

Accepted Answer

$$0.0918$$

Question 8

A GRE score is normally distributed with mean $$\mu = 1049$$ and standard deviation $$\sigma = 189. $$What score corresponds to the 85th percentile?

Accepted Answer

$$x = 1049 + 1.04 	imes 189 = 1246$$

Question 9

For a binomial experiment with $$n = 400$$ and $$p = 0.35$$, what is the mean and standard deviation?

Accepted Answer

$$\mu = 140$$, $$\sigma = 9.54$$

Question 10

Which condition must be met for the normal approximation to the binomial distribution to be valid?

Accepted Answer

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

Question 11

In a normal probability plot, what does an approximately linear pattern indicate?

Accepted Answer

The data is normally distributed.

Question 12

If the area under the normal curve between $$x_1$$ and $$x_2 is 0.90$$, what proportion of the population falls outside this interval?

Accepted Answer

0.10

Question 13

For a standard normal distribution, what is the probability that $$Z$$ falls between $$-1$$ and $$1$$?

Accepted Answer

0.6826

Question 14

What is the formula for converting a normal random variable $$X to a $$standard normal variable $$Z$$?

Accepted Answer

$$Z = \frac{X - \mu}{\sigma}$$

Question 15

A manufacturer wants to accept the middle 90% of steel rods. If the rod lengths are normally distributed with mean $$100 cm $$and standard deviation $$0.45 cm$$, what are the cutoff values?

Accepted Answer

Between $$99.26 cm $$and $$100.74 cm$$

Question 16

Which of the following is a correct interpretation of the area under a normal curve to the left of $$x$$?

Accepted Answer

It is the probability that a randomly selected value is less than $$x.$$

Question 17

If a binomial random variable $$X is $$approximated by a normal distribution, how should you adjust the interval for $$P(X = a)$$?

Accepted Answer

Use $$P(a - 0.5 \leq X \leq a + 0.5)$$

Question 18

What is the purpose of using a normal probability plot?

Accepted Answer

To assess whether sample data is approximately normal.

Question 19

If the area under the standard normal curve to the left of $$z = -0.38 is$$ $$0.3520$$, what is the area to the right of $$z = -0.38$$?

Accepted Answer

$$0.6480$$

The Normal Probability Distribution: Properties, Applications, and Approximations

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