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Chapter 6: Problem 72

Colorado has a high school graduation rate of \(75 \%\). a. In a random sample of 15 Colorado high school students, what is the probability that exactly 9 will graduate? b. In a random sample of 15 Colorado high school students, what is the probability that 8 or fewer will graduate? c. What is the probability that at least 9 high school students in our sample of 15 will graduate?

Short Answer

Expert verified
The probability that exactly 9 will graduate is given by plugging into the binomial distribution formula for part a. For part b, the probability 8 or fewer will graduate can be found by summing the probabilities for each number of students from 0 to 8. Using the complement rule, for part c, the probability that at least 9 will graduate is 1 minus the probability that 8 or fewer will graduate.

Step by step solution

01

Compute Probability for Part a

Use the binomial distribution formula, \(P(x; n, p) = \binom{n}{x} \cdot p^x \cdot (1-p)^{(n-x)}\), where n = 15, p = 0.75, and x = 9. This gives: \(P(9; 15, 0.75) = \binom{15}{9} \cdot (0.75)^9 \cdot (0.25)^6\)
02

Compute Probability for Part b

For part b, we need to calculate the probability for 8 or fewer students. Denote this as \(P(x \leq 8)\). This is: \(P(x \leq 8) = \sum_{x=0}^{8} \binom{15}{x} \cdot (0.75)^x \cdot (0.25)^{15-x}\)
03

Compute Probability for Part c

For part c, we want to find \(P(x \geq 9)\), the probability that at least 9 students will graduate. We could add up the probabilities for \(x = 9, 10, ..., 15\), but it鈥檚 easier to use the complement rule: \(P(x \geq 9) = 1 - P(x \leq 8)\)

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Most popular questions from this chapter

Rule with z-Scores The Empirical Rule applies rough approximations to probabilities for any unimodal, symmetric distribution. But for the Normal distribution we can be more precise. Use the figure and the fact that the Normal curve is symmetric to answer the questions. Do not use a Normal table or technology. According to the Empirical Rule, a. Roughly what percentage of \(z\) -scores are between \(-2\) and 2 ? i. almost all iii. \(68 \%\) ii. \(95 \%\) iv. \(50 \%\) b. Roughly what percentage of \(z\) -scores are between \(-3\) and 3 ? i. almost all iii. \(68 \%\) ii. \(95 \%\) iv. \(50 \%\) c. Roughly what percentage of \(z\) -scores are between \(-1\) and 1 . i. almost all iii. \(68 \%\) ii. \(95 \%\) iv. \(50 \%\) d. Roughly what percentage of \(z\) -scores are greater than \(0 ?\) i. almost all iii. \(68 \%\) ii. \(95 \%\) iv. \(50 \%\) e. Roughly what percentage of \(z\) -scores are between 1 and \(2 ?\) i. almost all iii. \(50 \%\) ii. \(13.5 \%\) iv. \(2 \%\)

Toronto drivers have been going to small towns in Ontario (Canada) to take the drivers' road test, rather than taking the test in Toronto, because the pass rate in the small towns is \(90 \%\), which is much higher than the pass rate in Toronto. Suppose that every day, 100 people independently take the test in one of these small towns. a. What is the number of people who are expected to pass? b. What is the standard deviation for the number expected to pass? c. After a great many days, according to the Empirical Rule, on about 95\% of these days the number of people passing the test will be as low as _______ and as high as _______. d. If you found that on one day, 89 out of 100 passed the test, would you consider this to be a very high number?

The weight of newborn hippopotami is approximately Normal, with a mean of 88 pounds and a standard deviation of 10 pounds. a. What is the probability that a newborn hippo weighs between 90 and 110 pounds? b. Suppose baby hippos that weigh at the 5 th percentile or less at birth are unlikely to survive. What weight corresponds with the 5 th percentile for newborn hippos? c. Fiona the Hippo was born at the Cincinnati Zoo in 2017,6 weeks premature, and weighed only 29 pounds at birth. What percentage of baby hippos are born weighing 29 pounds or less?

Whales have one of the longest gestation periods of any mammal. According to whalefacts.org, the mean gestation period for a whale is 14 months. Assume the distribution of gestation periods is Normal with a standard deviation of \(1.2\) months. a. Find the standard score associated with a gestational period of \(12.8\) months. b. Using the Empirical Rule and your answer to part a, what percentage of whale pregnancies will have a gestation period between \(12.8\) months and 14 months? c. Would it be unusual for a whale to have a gestation period of 18 months? Why or why not?

According to the American Veterinary Medical Association, \(30 \%\) of Americans own a cat. a. Find the probability that exactly 2 out of 8 randomly selected Americans own a cat. b. In a random sample of 8 Americans, find the probability that more than 3 own a cat.
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