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