Q: A restaurant’s daily sales are normally distributed with a mean of €2100 and a standard deviation of €250. What is the upper control limit (UCL) for a control chart using 3 standard deviations?

$$UCL = 2100 + 3 \times 250 = 2850$$

Q: Suppose the weekly hours worked by junior doctors at a hospital are normally distributed with a mean of 45 hours and a standard deviation of 5 hours. What is the number of hours that only 5% of junior doctors exceed?

$$45 + 1.645 \times 5 = 53.225$$ hours

Question 1

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

Accepted Answer

The total area under the curve is less than 1.

Question 2

Suppose the heights of adult females are normally distributed with a mean of 170 cm and a standard deviation of 6 cm. What is the z-score for a woman who is 182 cm tall?

Accepted Answer

$$z = \frac{182 - 170}{6} = 2$$

Question 3

If IQ scores are normally distributed with a mean ($$\mu) of 100 $$and a standard deviation ($$\sigma) of 15$$, what percentage of people have an IQ between 85 and 115?

Accepted Answer

About 68%

Question 4

A hospital finds that the weekly hours worked by junior doctors are normally distributed with a mean of 45 hours and a standard deviation of 5 hours. What is the probability that a randomly selected junior doctor works more than 50 hours per week?

Accepted Answer

About 15.87%

Question 5

Which formula is used to transform a value $$X$$ from any normal distribution to a standard normal variable $$Z$$?

Accepted Answer

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

Question 6

A restaurant’s daily sales are normally distributed with a mean of €2100 and a standard deviation of €250. What is the upper control limit (UCL) for a control chart using 3 standard deviations?

Accepted Answer

$$UCL = 2100 + 3 	imes 250 = 2850$$

Question 7

If the area under the normal curve between $$z = -1$$ and $$z = 1 is $$approximately 0.6826, what does this represent?

Accepted Answer

About 68% of the data falls within one standard deviation of the mean.

Question 8

Suppose the birth weights of thoroughbred foals are normally distributed with a mean of 54 kg and a standard deviation of 6.3 kg. What is the z-score for a foal weighing 65 kg?

Accepted Answer

$$z = \frac{65 - 54}{6.3} \approx 1.75$$

Question 9

A patient’s diastolic blood pressure is measured at 125 mm Hg. If the population mean is 83 mm Hg with a standard deviation of 10 mm Hg, what is the z-score for this patient?

Accepted Answer

$$z = \frac{125 - 83}{10} = 4.2$$

Question 10

Which of the following statements about continuous random variables is TRUE?

Accepted Answer

The probability that a continuous random variable takes any exact value is zero.

Question 11

If 99.7% of observations in a normal distribution fall within 3 standard deviations of the mean, what is the probability that an observation falls outside this range?

Accepted Answer

0.3%

Question 12

A company uses a control chart to monitor daily sales, which are normally distributed with a mean of €2100 and a standard deviation of €250. On which day(s) is the sales figure considered an unusual observation if the sales are: 2300, 1850, 2300, 1600, 2900, 2400?

Accepted Answer

Day 5 only

Question 13

Which of the following best describes the use of z-values in employee appraisals for large organizations?

Accepted Answer

Z-values can be used to compare employees’ performance relative to the group.

Question 14

If the standard deviation of a normal distribution increases while the mean remains the same, what happens to the shape of the curve?

Accepted Answer

The curve becomes wider and flatter.

Question 15

Suppose the weekly hours worked by junior doctors at a hospital are normally distributed with a mean of 45 hours and a standard deviation of 5 hours. What is the number of hours that only 5% of junior doctors exceed?

Accepted Answer

$$45 + 1.645 	imes 5 = 53.225$$ hours

Question 16

Which of the following is a real-world application of the normal distribution?

Accepted Answer

Quality control using control charts in manufacturing.

Question 17

If $$P(Z \u003e 1.33) = 0.0918$$, what does this mean in the context of a standard normal distribution?

Accepted Answer

9.18% of values are greater than 1.33 standard deviations above the mean.

Normal Probability Distribution: Concepts, Calculations, and Applications

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