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Chapter 7
(Discrete Probability Distributions): Exercise 7.97
7.97 In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 10% of all voters are Independent. A survey asked 25 people to identify themselves as Democrat, Republican, or Independent.
a. What is the probability that none of the people are Independent?
b. What is the probability that fewer than five people are Independent?
c. What is the probability that more than two people are Independent?
|Formula is P(x) = C(n, x) p^x q^(n - x) |
|Here, n = 25, p = 0.1, q = 1 - p = 0.9 |
|(a) P(0) = C(25, 0) 0.1^0 0.9^(25 - 0) |
|= 1 * 1 * 0.3487 |
|= 0.0718 |
|(b) P(Fewer than 5) = P(≤ 4) = P(0) + P(1) + … +P(4) |
|= C(25, 0) 0.1^0 0.9^(25 - 0) + C(25, 1) 0.1^1 0.9^(25 - 1) + … + C(25, 4) 0.1^4 0.9^(25 - 4) |
|= 0.0718 + 0.1994 + 0.2659 + 0.2265 + 0.1384 |
|= 0.9020 |
|(c) P(More than 2) = 1 - P(2 or less) = 1 - (x ≤ 2) = 1 - [P(0) + P(1) + P(2)] |
|= 1 - [C(25, 0) 0.1^0 0.9^(25 - 0) + C(25, 1) 0.1^1 0.9^(25 - 1) + C(25, 2) 0.1^2 0.9^(25 - 2)] |
|= 1 - (0.0718 + 0.1994 + 0.2659) |
|= 0.4629 |
Chapter 8
(Continuous Probability Distributions): Exercise 8.42
8.42. Travelbus is an Internet-based travel agency wherein customers can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400.
a. What is the probability of getting more than 12,000 hits?
b. What is the probability of getting fewer than 9,000 hits?
| μ = 10000, σ = 2400 |
|(a) x = 12000 |
|z = (x - μ)/σ = (12000 - 10000)/2400 = 0.8333 |
|P(x > 12000) = P(z > 0.8333) = 0.2023 |
|(b) x = 9000 |
|z = (x - μ)/σ = (9000 - 10000)/2400 = -0.4167 |
|P(x < 9000) = P(z < -0.4167) = 0.3385 |
Chapter 13
(Inference about Comparing two Populations): Exercise 13.8
13.8 Xr13-08 A men's softball league is experimenting with a yellow baseball that is easier to see during night games. One way to judge the effectiveness is to count the number of errors. In a preliminary experiment, the yellow baseball was used in 10 games and the traditional white baseball was used in another 10 games. The number of errors in each game was recorded and is listed here. Can we infer that there are fewer errors on average when the yellow ball is used?
|Yellow |
|White |
| |
|5 |
|7 |
| |
|2 |
|6 |
| |
|6 |
|8 |
| |
|7 |
|5 |
| |
|2 |
|9 |
| |
|5 |
|11 |
| |
|3 |
|8 |
| |
|8 |
|3 |
| |
|4 |
|6 |
| |
|9 |
|10 |
| |
| |
|n1 = 10 |
|n2 = 10 |
|x1-bar = 5.1 |
|x2-bar = 7.3 |
|s1 = 2.42 |
|s2 = 2.41 |
|Hypotheses: |
|Ho: μ1 ≥ μ2 |
|Ha: μ1 < μ2 |
|Decision Rule: |
|α = 0.05 |
|Degrees of freedom = 10 + 10 - 2 = 18 |
|Critical t- score = -1.734063592 |
|Reject Ho if t < -1.734063592 |
|Test Statistic: |
|Pooled SD, s = √ {(n1 - 1) s1^2 + (n2 - 1) s2^2} / (n1 + n2 - 2)] = √(((10 - 1) * 2.42^2 + (10 - 1) * 2.41^2) / (10 + 10 - 2)) = 2.415005176 |
|SE = s * √{(1 /n1) + (1 /n2)} = 2.41500517597789 * √((1/10) + (1/10)) = 1.080023148 |
|t = (x1-bar -x2-bar)/SE = (5.1 - 7.3)/1.08002314790008 = -2.036993378 |
|p- value = 0.02831554 |
|Decision (in terms of the hypotheses): |
|Since -2.036993378 < -1.734063592 we reject Ho and accept Ha |
|Conclusion (in terms of the problem): |
|It appears that there are fewer errors on average when the yellow ball is used. |
Chapter 15
(Chi Square Tests): Exercise 15.48
15.48 Xr15-48 Every week the Florida Lottery draws six numbers between 1 and 49. Lottery ticket buyers are naturally interested in whether certain numbers are drawn more frequently than others. To assist players, the Sun-Sentinel publishes the number of times each of the 49 numbers has been drawn in the past 52 weeks. The numbers and the frequency with which each occurred were recorded. (These data are from the Sunday, January 5, 1997, edition.)
a. If the numbers are drawn from a uniform distribution, what is the expected frequency for each number?
b. Can we infer that the data were not generated from a uniform distribution?
|Number Drawn |
|Observed Frequency (fo) |
|Expected Frequency Per Uniform Distribution (fe) |
|(fo - fe)^2 /fe |
| |
|3 |
|3 |
|6 |
|1.50 |
| |
|8 |
|5 |
|6 |
|0.17 |
| |
|12 |
|3 |
|6 |
|1.50 |
| |
|36 |
|10 |
|6 |
|2.67 |
| |
|44 |
|15 |
|6 |
|13.50 |
| |
|45 |
|7 |
|6 |
|0.17 |
| |
| |
| |
|χ2 = |
|19.50 |
| |
| |
|(a) Expected frequency for each number = 6 |
|(b) Ho: The frequency of winning numbers follows a uniform distribution |
|Ha: The frequency of winning numbers does not follow a uniform distribution |
|Test χ2 value = 19.50 |
|p- value (dof = 5) is 0.0016 |
|Since 0.0016 < 0.05, we reject Ho and accept Ha |
|Conclusion: It appears that the frequency of winning numbers does not follow a uniform distribution |
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