# Number of bottles and the probability of each number

Please see the attached file for full problem description.

1. Given the number of bottles (0,1,2,3,4) and the probability of each number being overfilled in one month, what is the expected number of overfilled bottles per month?

E(X) = SUM(x*P(x))

2. Given n=4 & p=.3 for a binomial experiment, the probability of x=0 is?

3. The height of basketball players is normally distributed with a mean of 79 inches, and std dev of 3 inches. What is the probability of a player having a height less than 73 inches?

4. You're given 4 bins and the frequency for each bin (Grouped Data). What is the mean value? What is the % cumulative probability that the frequency falls within the first two bins? See Class Notes on Grouped Data. Recall that mean=ΣfM/Σf

5. A population is normally distributed. What proportion will fall within Z1 & Z2? Use the Z tables to get P(Z1<Z<Z2).

6. What is the probability of A or B if P(A), P(B) & P(A&B) can be computed from the stated question? What is the P(A&B) if these events are dependent, P(A&B)=P(A)P(B|A), and the data is presented in a two way table?

7. Know the range of the coefficient of correlation, r, and the difference between it and the coefficient of determination, r2. What do these properties mean?

8. 25% of students in a class of 60 are going to grad school. The mean and std dev of this binomial distribution is ?

9. Given the critical and test values of Z for a hypothesis test, what is the p-value?

10. Given the probability of an event in an experiment which is performed twice without success, then on the third trial it: must occur, may occur, could not occur, has a probability of that involves it's probability raised to some power.

11. The length of parts produced is normally distributed with a mean of 10 inches and a std dev of 2 inches. What percentage will be at least 13.7 inches. P(X>13.7)

12. Given that salaries for football players is right (positive) skewed, should the players union argue for increases based on the mean or median salary?

13. If 70% of students are female, then for a sample of 10 students what is the probability that 0 are female.

14. For #13, what is the probability that X≥ 1.

15. For a normal variable, what is the value of Z if the area to the left of Z is .9000.

16. Given a hypothesis test to be performed, select the appropriate statement of Ho & Ha.

17. You are given an ASP software print out for a "Compare Two Means: Independent Samples" for a small sample size and ask which assumption is not necessary for a valid test. Don't freak out and think "I've never used the ASP software." The print out is a smoke screen. Just think about the assumptions underlying this type of test.

18. If a hypothesis test is conducted for alpha = 0.10, what p-values would lead to the null hypothesis being rejected.

19. What are the assumptions for using ANOVA to compare 3 or more population means? Given the computer output for an ANOVA analysis, what is the critical value and should Ho be rejected?

20. The observed frequency (or %) is given for k categories. Perform a Chi-Square Goodness of Fit test for both equal & unequal expected frequencies.

21. There are four T/F questions regarding: independent vs mutually exclusive events, Type I errors, properties of the Chi-Square distribution, and the drawback of some descriptive graph presentations as related to measure of reliability.

22. A market has a monthly demand for a certain product that is normally distributed with a mean of 100 and std dev of 25. How many of this product must be in inventory for a month so that only 1.5% chance that all will be sold?

23. The amount of liquid dispensed into a 12 oz bottle has a normal distribution with a mean of 12.7 ounces and a std dev of .3 ounces. If 100 bottles are randomly selected, what is the probability that the sample mean exceeds 12.4 ounces.

24. Use Chebychev's Rule to compute the maximum expected percent of the random variable that will be above a certain value of the std dev. Recall that at least (1-1/k^2) will be the fraction of the values that fall within ± k (std dev) of the mean. So, no more than 1-(1-1/k^2) can be greater than (1-1/k^2).

25. An agency employs 50 workers who interview applicants for loans. The boss randomly selects loans processed by two workers to check for errors. 5 of the workers in the group have consistently made errors. What is the probability that both workers checked have made errors? P(worker makes errors)=5/50. Compute P(x=2) from binomial equation using combination of 50 taken two at a time and p=.1.

#### Solution Summary

The number of bottles and the probability of each number are determined.