Question 1

Suppose the annual cost per employee for dental insurance at a company is normally distributed with a mean of $$1,520 and a standard deviation of 420. $$What is the probability that a randomly selected employee has a cost less than $1,280?

Accepted Answer

$$P(X \u003c 1280) = P\left(Z \u003c \frac{1280 - 1520}{420}\right) = P(Z \u003c -0.57) = 0.2843$$

Question 2

A local business has an average of 56 customers per day with a standard deviation of 8. What is the probability that on a randomly selected day, the business has at least 64 customers?

Accepted Answer

$$P(X \geq 64) = P\left(Z \geq \frac{64 - 56}{8}\right) = P(Z \geq 1) = 0.1587$$

Question 3

Which of the following best describes the empirical rule for a normal distribution?

Accepted Answer

Approximately 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3 standard deviations from the mean.

Question 4

If $$Z is a $$standard normal variable, what is $$P(-1.96 \u003c Z \u003c 1.96)$$?

Accepted Answer

$$0.9500$$

Question 5

A manager’s salary is normally distributed with a mean of $$54,000 and a standard deviation of 2$$,000. What is the probability that a randomly selected manager earns more than $56,500?

Accepted Answer

$$P(X \u003e 56500) = P\left(Z \u003e \frac{56500 - 54000}{2000}\right) = P(Z \u003e 1.25) = 0.1056$$

Question 6

Which theorem guarantees that at least $$1 - \frac{1}{k^2$} of $$the data lies within $$k$$ standard deviations of the mean for any distribution?

Accepted Answer

Chebyshev’s Theorem

Question 7

Given $$Z is a $$standard normal variable, what is $$P(Z \u003e 2.33)$$?

Accepted Answer

$$0.0099$$

Question 8

If the probability that $$Z is $$between $$-1$$ and $$1 is$$ $$0.6826$$, what is the probability that $$Z is $$outside this range?

Accepted Answer

$$1 - 0.6826 = 0.3174$$

Question 9

A company finds that the annual cost per employee for dental insurance is normally distributed with mean $$\mu = 1520$$ and $$\sigma = 420. $$What is the $$Z-$$score for an employee whose cost is $$2000$$?

Accepted Answer

$$Z = \frac{2000 - 1520}{420} = 1.14$$

Question 10

If $$P(Z \u003c a) = 0.9750$$ for a standard normal variable $$Z$$, what is the value of $$a$$?

Accepted Answer

$$a = 1.96$$

Question 11

What is the probability that a standard normal variable $$Z is $$between $$-2$$ and $$2$$?

Accepted Answer

$$P(-2 \u003c Z \u003c 2) = 0.9544$$

Question 12

If the mean number of customers per day is $$56$$ with a standard deviation of $$8$$, what is the $$Z-$$score for a day with $$44$$ customers?

Accepted Answer

$$Z = \frac{44 - 56}{8} = -1.5$$

Question 13

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

Accepted Answer

It is skewed to the right.

Question 14

If $$P(Z  1.28)$$?

Accepted Answer

$$1 - 0.8997 = 0.1003$$

Question 15

A random variable $$X is $$normally distributed with mean $$\mu = 56$$ and $$\sigma = 8. $$What is the probability that $$X is $$between $$48$$ and $$64$$?

Accepted Answer

$$P(48 \u003c X \u003c 64) = P(-1 \u003c Z \u003c 1) = 0.6826$$

Step-by-Step Guidance for Normal Distribution and Z-Score Problems

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