# Exercises - Normal Distributions

1. Find $P(z \lt 1.23)$, the area under the standard normal curve to the left of $z=1.23$

2. Find $P(-0.67 \lt z \lt 0)$, the area under the standard normal curve between $z=-0.67$ and $z=0$.

3. Find each of the following probabilities

1. $P(z \gt 2.30)$
2. $P(-0.33 \lt z \lt 1.45)$
3. $P(z \gt -2.1)$
4. $P(z \lt -3.5)$

4. Find the number $b$ that satisfies:

1. $P(z \lt b) = 0.9664$
2. $P(z \lt b) = 0.3050$
3. $P(z \gt b) = 0.0078$

5. Find $Q_1$, $Q_2$, and $Q_3$ on the standard normal distribution.

6. Find the 90th percentile on the standard normal distribution.

7. Find $P(-1 \lt z \lt 1)$ and $P(-2 \lt z \lt 2)$. Does your answer agree with the Empirical Rule?

8. The average salary for first-year teachers is $\$27,989$. If the distribution is approximately normal with standard deviation$\$3250$,

1. what is the probability that a randomly selected first-year teacher makes between $\$20,000$and$\$30,000$ each year?

2. what is the probability that a randomly selected first-year teacher has a salary less than $\$20,000$? 9. The national average SAT score is 1019. If we assume a normal distribution with standard deviation 90, 1. what is the probability that a randomly selected score exceeds 1200? 2. what is the 90th percentile score? 10. The average time a person spends at the Barefoot Landing Seaquarium is 96 minutes. The standard deviation is 17 minutes. Assume the variable is normally distributed. 1. If a visitor is selected at random, find the probability that he or she will spend at least 120 minutes at the seaquarium. 2. If a visitor is selected at random, find the probability that he or she will spend at most 80 minutes at the sequarium. 3. Suggest a time for a bus to return to pick up a group of tourists 11. The average charitable contribution itemized per income tax return in Pennsylvania is$\$792$. Suppose that the distribution of contributions is normal with a standard deviation of $\$103\$. Find the limits for the middle 50% of contributions.

12. Is the following data set normal? $$\begin{array}{cccccccc} 3 & 58 & 5 & 65 & 17 & 48 & 52 & 75\\ 21 & 76 & 58 & 36 & 100 & 111 & 34 & 41\\ 23 & 44 & 33 & 50 & 13 & 18 & 7 & 12\\ 20 & 24 & 66 & 28 & 28 & 31 & & \end{array}$$