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?

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

Question 1

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

Accepted Answer

The curve is skewed to the right.

Question 2

Suppose the heights of adult females are normally distributed with a mean of 165 cm and a standard deviation of 6 cm. What is the z-score for a woman who is 177 cm tall?

Accepted Answer

$$z = \frac{177 - 165}{6} = 2$$

Question 3

If IQ scores are normally distributed with $$\mu = 100$$ and $$\sigma = 15$$, 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 from any normal distribution to the standard normal distribution?

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?

Accepted Answer

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

Question 7

If the area under the normal curve between $$\mu - 1\sigma$$ and $$\mu + 1\sigma is $$approximately 0.6826, what is the probability that a value falls outside this range?

Accepted Answer

0.3174

Question 8

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 patient’s z-score?

Accepted Answer

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

Question 9

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 10

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 percentage of foals have a birth weight above 65 kg?

Accepted Answer

About 2.5%

Question 11

Which real-world application is NOT typically associated with the normal distribution?

Accepted Answer

Counting the number of cars passing a checkpoint in an hour.

Question 12

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

Accepted Answer

0.003

Question 13

A company uses a forced ranking system for employee appraisals based on a normal distribution. If the top 20% of employees are eligible for promotion, what is the approximate z-score cutoff for promotion eligibility?

Accepted Answer

$$z \approx 0.84$$

Question 14

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

Accepted Answer

The curve becomes wider and flatter.

Question 15

If the probability that $$Z \u003e 1.33 is 0.0918$$, what is the probability that $$Z \u003c 1.33$$?

Accepted Answer

0.9082

Question 16

A control chart for daily sales uses $$\mu = 2100$$ and $$\sigma = 250. If a $$day’s sales are €2900, how many standard deviations above the mean is this value?

Accepted Answer

$$z = \frac{2900 - 2100}{250} = 3.2$$

Question 17

Which of the following is a correct interpretation of the area under the normal curve between $$z = -1$$ and $$z = 1$$?

Accepted Answer

It represents the probability that a value falls within one standard deviation of the mean.

L8 Normal Probability Distribution: Concepts, Calculations, and Applications

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