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# Binomial Random Variables, Probability, and Normal Distribution

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7.84) Given a binomial random variable with n = 10 and p = .3, use the formula to find the following probabilities.
a. P(X = 3)
b. P(X = 5)
c. P(X = 8)

7.97) In the United States, voters who are neither Democrat nor Republican are called Independents. It is believed that 10% of all voters are Independents. 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?

8.35)
X is normally distributed with mean 250 and standard deviation 40. What value of X does only the top 15% exceed?

8.42)
Travelbyus 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?

Exercises 13.5 to 13.10 are "what-if" analyses designed to determine what happens to the test statistics and interval estimates when elements of the statistical inference change. These problems can be solved manually, using the Excel spreadsheets you created or Minitab.

13.5) In random samples of 25 from each of two normal populations, we found the following statistics:

a. Estimate the difference between the two population means with 95% confidence.
b. Repeat part (a) increasing the standard deviations to s1 = 255 and s2 = 260.
c. Describe what happens when the sample standard deviations get larger.
d. Repeat part (a) with samples of size 100.
e. Discuss the effects of increasing the sample size.

13.8) Random sampling from two normal populations produced the following results:
X1 = 412 S1 = 128 n1 = 150
X2 = 405 S2 = 54 n2 = 150
a. Can we infer at the 5% significance level that μ1 is greater than μ2?
b. Repeat part (a) decreasing the standard deviations to s1 = 31 and s2 = 16.
c. Describe what happens when the sample standard deviations get smaller.
d. Repeat part (a) with samples of size 20.
e. Discuss the effects of decreasing the sample size.
f. Repeat part (a) changing the mean of sample 1 to x1 = 409
g. Discuss the effect of decreasing x1

15.48) A random sample of 50 observations yielded the following frequencies for the standardized intervals:
Interval Frequency
z ≤ - 1 6
- 1 < z ≤ 0 27
0 < z ≤ 1 14
z > 1 3

Can we infer that the data are not normal? (Use α = .10.)
The following exercises require the use of a computer and software.

15.56) Suppose that the personnel department in Exercise 15.55 continued its investigation by categorizing absentees according to the shift on which they worked, as shown in the accompanying table. Is there sufficient evidence at the 10% significance level of a relationship between the days on which employees are absent and the shift on which the employees work?

Shift Monday Tuesday Wednesday Thursday Friday
Day 52 28 37 31 33
Evening 35 34 34 37 41