Question 1

Suppose the heights of adult males in a population are normally distributed with mean $$\mu = 70$$ inches and standard deviation $$\sigma = 3$$ inches. What is the probability that a randomly selected male is taller than 76 inches?

Accepted Answer

$$P(X \u003e 76) = 1 - P(Z \u003c 2) = 1 - 0.9772 = 0.0228$$

Question 2

A pharmaceutical company claims that its drug increases the average recovery rate. The recovery times are normally distributed with $$\mu = 8$$ days and $$\sigma = 2$$ days. What is the z-score for a patient who recovers in 5 days?

Accepted Answer

$$z = \frac{5 - 8}{2} = -1.5$$

Question 3

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

Accepted Answer

It has a mean of $$0$$ and a standard deviation of $$1.$$

Question 4

A researcher wants to find the probability that a randomly selected value from a normal distribution with $$\mu = 50$$ and $$\sigma = 10 is $$between 45 and 65. What is the correct approach?

Accepted Answer

Convert both values to z-scores and use the standard normal table: $$z_1$$ = \frac{45-50}{10} = -0.5$$, z_2$ = \frac{65-50}{10} = 1.5$$; then $$P(45 \u003c X \u003c 65) = P(-0.5 \u003c Z \u003c 1.5).$$

Question 5

If $$X \sim N(100, 16)$$, what is the probability that $$X is $$less than 92?

Accepted Answer

$$z = \frac{92-100}{4} = -2$$, so $$P(X \u003c 92) = P(Z \u003c -2) = 0.0228$$

Question 6

Which property is NOT true for all normal distributions?

Accepted Answer

The standard deviation is always 1.

Question 7

A test score is normally distributed with $$\mu = 80$$ and $$\sigma = 5. $$What score corresponds to the 90th percentile?

Accepted Answer

Find $$z$$ for 90th percentile ($$z \approx 1.28$$): $$X = 80 + 1.28 	imes 5 = 86.4$$

Question 8

Which scenario best illustrates the use of the empirical rule (68-95-99.7 rule)?

Accepted Answer

Estimating the percentage of students scoring within one standard deviation of the mean on a normally distributed exam.

Question 9

A hospital records the weights of newborns, which are normally distributed with $$\mu = 3.5 kg $$and $$\sigma = 0.5 kg. $$What proportion of newborns weigh between 3.0 kg and 4.0 kg?

Accepted Answer

Convert to z-scores: $$z_1$$ = \frac{3.0-3.5}{0.5} = -1$$, z_2$ = \frac{4.0-3.5}{0.5} = 1$$; $$P(-1 \u003c Z \u003c 1) = 0.6826$$

Question 10

Which of the following is the formula for converting a raw score $$X to a z-$$score in a normal distribution?

Accepted Answer

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

Question 11

A company finds that the time to assemble a product is normally distributed with mean 30 minutes and standard deviation 4 minutes. What is the probability that a randomly selected assembly takes between 26 and 34 minutes?

Accepted Answer

Convert to z-scores: $$z_1$$ = \frac{26-30}{4} = -1$$, z_2$ = \frac{34-30}{4} = 1$$; $$P(-1 \u003c Z \u003c 1) = 0.6826$$

Question 12

Which of the following best describes the effect of increasing the standard deviation $$\sigma in a $$normal distribution?

Accepted Answer

The distribution becomes wider and flatter.

Question 13

A student scores 1.5 standard deviations above the mean on a standardized test. What percentage of students scored lower than this student?

Accepted Answer

$$P(Z \u003c 1.5) = 0.9332$$, so 93.32\% scored lower.

Question 14

If the area to the left of $$z = 0.75 is 0.7734$$, what is the area to the right of $$z = 0.75$$?

Accepted Answer

$$1 - 0.7734 = 0.2266$$

Question 15

A population of exam scores is normally distributed with $$\mu = 60$$ and $$\sigma = 8. $$What score marks the top 5\% of the distribution?

Accepted Answer

Find $$z$$ for top 5\% ($$z \approx 1.645$$): $$X = 60 + 1.645 	imes 8 = 73.16$$

Question 16

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

Accepted Answer

It has two modes.

Question 17

A sample of blood pressure readings is normally distributed with $$\mu = 120$$ mmHg and $$\sigma = 10$$ mmHg. What is the probability that a reading is less than 110 mmHg?

Accepted Answer

$$z = \frac{110-120}{10} = -1$$, so $$P(Z \u003c -1) = 0.1587$$

Question 18

Which of the following best explains why the normal distribution is important in statistics?

Accepted Answer

Many natural phenomena are approximately normally distributed, and it is the basis for many statistical inference methods.

Normal Distribution and Z-Scores: Study Notes for Statistics

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