Exam 5: Continuous Random Variables

Which one of the following suggests that the data set is approximately normal?

Suppose x is a random variable best described by a uniform probability distribution with c = 20 and d = 60. Find P(x > 52).

The following data represent the scores of a sample of 50 students on a statistics exam. The mean score is $\bar { x } = 80.3$ , and the standard deviation is $s = 11.37$ . 49 51 59 63 66 68 68 69 70 71 71 71 73 74 76 76 76 77 78 79 79 79 79 80 80 82 83 83 83 85 85 86 86 88 88 88 88 89 89 89 90 91 92 92 93 95 96 97 97 98 What percentage of the scores fall in each of the intervals $\bar { x } \pm \mathrm { s } , \bar { x } \pm 2 \mathrm {~s}$ , and $\bar { x } \pm 3 \mathrm {~s}$ ? Based on these percentages, do you believe that the distribution of scores is approximately normal? Explain.

For a standard normal random variable, find the probability that z exceeds the value -1.65.

The amount of soda a dispensing machine pours into a 12-ounce can of soda follows a normal distribution with a mean of 12.54 ounces and a standard deviation of 0.36 ounce. Each can holds a maximum of 12.90 ounces of soda. Every can that has more than 12.90 ounces of soda poured into it causes a spill and the can must go through a special cleaning process before it can be sold. What is the probability that a randomly selected can will need to go through this process?

A machine is set to pump cleanser into a process at the rate of 9 gallons per minute. Upon inspection, it is learned that the machine actually pumps cleanser at a rate described by the uniform distribution over the interval 8.5 to 11.5 gallons per minute. Find the probability that between 9.0 gallons and 10.0 gallons are pumped during a randomly selected minute.

Which shape is used to represent areas for a normal distribution?

Suppose x is a random variable best described by a uniform probability distribution with c = 3 and d = 9. Find the value of a that makes the following probability statement true: P(3.5 ≤ x ≤ a) = 0.5.

Suppose that the random variable x has an exponential distribution with θ = 1.5. Find the probability that x will assume a value within the interval μ ± 2σ.

A certain baseball player hits a home run in 4% of his at-bats. Consider his at-bats as independent events. Find the probability that this baseball player hits at most 16 home runs in 650 at-bats?

The age of customers at a local hardware store follows a uniform distribution over the interval from 18 to 60 years old. Find the average age of customers to this hardware store.

Suppose x is a random variable best described by a uniform probability distribution with c = 10 and d = 50. Find P(40 ≤ x ≤ 50).

Suppose x is a uniform random variable with c = 20 and d = 50. Find P(23 < x < 45). Round to the nearest hundredth when necessary.

Use the standard normal distribution to find $P ( - 2.25 < z < 0 )$

Assume that x is a binomial random variable with n = 400 and p = 0.30. Use a normal approximation to find P(x > 100).

A paint machine dispenses dye into paint cans to create different shades of paint. The amount of dye dispensed into a can is known to have a normal distribution with a mean of 5 milliliters (ml) and a standard deviation of 0.4 ml. Answer the following questions based on this information. Find the dye amount that represents the 9th percentile of the distribution.

P(-1 < x < 0) = P(0 < x < 1) for any random variable x that is normally distributed.

A certain baseball player hits a home run in 6% of his at-bats. Consider his at-bats as independent events. Find the probability that this baseball player hits more than 32 home runs in 750 at-bats?

For a standard normal random variable, find the point in the distribution in which 11.9% of the z -values fall below.

Suppose x is a random variable best described by a uniform probability distribution with c = 2 and d = 6. Find the value of a that makes the following probability statement true: P(x ≤ a) = 1.