# Critical thinking: you are given the information that

6. Critical thinking: You are given the information that P (A)=0.30 and P(B)=0.40.

(a) Do you have enough information to compute P (A and B)? Explain.

(b) If you know that events A and B are mutually exclusive, do you have enough information to compute P (A and B)? Explain.

8. Critical thinking: for a class activity, your group has been assigned the task of generating a quiz question that requires use of the formula for conditional probability to compute P (A|B). Your group comes up with the following question: “If P (A and B)=0.40 and P (A)=0.20, what is the value of P (B|A)? what is wrong with this question? Hint: Consider the answer you get when using the correct formula, P (B|A)= P (A and B)/P (A).

12. Survey: Reaction to Poison Ivy. Allergic reactions to poison ivy can be miserable. Plant oils cause the reaction. Researchers at the Allergies Institute did a study to determine the effects of washing the oil off within 5 minutes of exposure. A random sample of 1000 people with known allergies to poison ivy participated in the study. Oil from the poison ivy plant was rubbed on a patch of skin. For 500 of the subjects, it was washed off within 5 minutes. For the other 500 subjects, the oil was washed off after 5 minutes. The results are summarized.

Reaction |
Within 5 Mins. |
After 5 Mins. |
Row Total |

None |
420 |
50 |
470 |

Mild |
60 |
330 |
390 |

Strong |
20 |
120 |
140 |

Column Total |
500 |
500 |
1000 |

Let’s use the following notation for the various events: W= washing oil off within 5 minutes, A= washing off after 5 minutes, N= no reaction, M= mild reactions, S= strong reaction. Find the following probabilities for a person selected at random from this sample of 1000 subjects.

(A). P (N), P (M), P (S)

(B). P (N|W), P (S|W)

(C). P (N|A), P (S|A)

(D). P (N and W), P (M and W)

(E ). P (N or M). Are the events N=no reaction and M-mild reaction mutually exclusive? Explain.

(F). Are the events N=no reaction and W= washing oil off within 5 minutes independent? Explain.

Part 2

2. Statistical literacy: List the criteria for a binomial experiment. What does the random variable of a binomial experiment measure?

4. Critical thinking: Consider a binomial experiment. If the number of trials is increased, what happens to the expected value? to the standard deviation? Explain.

8. Quality Control : Pens. A stationary store has decided to accept a large shipment of ball point pens if an inspection of 20 randomly selected pens yields no more than two defective pens.

(A). Find the probability that this shipment is accepted if 5% of the total shipment is defective.

(B.) Find the probability that this shipment is not accepted if 15% of the total shipment is defective.

Part 3

2. According to the empirical rule, approximately what percentage of the area under a normal distribution lies within 1 standard deviation of the mean? Within 2 standard deviations? Within 3 deviations?

4. Can a normal distribution always be used to approximate a binomial distribution? Explain.

10. If x has a normal distribution with mean u=15 and standard deviation o=3, describe the distribution of x values for sample size n, where n=4, n=16, and n=100. How do the x distribution compare for the various sample sizes?

12. Given that x is a normal variable with mean u= 110 and standard deviation o=12, find:

Part 4

2. Suppose you are told that a 95% confidence interval for the average price of a gallon of regular gasoline in your state is from $3.15 to $3.45. Use the fact that the confidence interval for the mean has the form X-E to X+E to compute the sample mean and the maximal margin of error E.

4. Anystate Auto Insurance Company took a random sample of 370 insurance claims paid out during a 1-year period. The average claim paid was $1570. Assume o=$250. Find 0.90 and 0.99 confidence intervals for the mean claim payment.

8. Shards of clay vessel were put together to reconstruct rim diameters of the original ceramic vessels found at the Wind Mountain archaeological site. A random sample of ceramic vessels gave the following rim diameters (in centimeters):

15.9, 13.4, 22.1, 12.7, 13.1, 19.6, 11.7, 13.5, 17.7, 18.1.

(A). Use a calculator with mean and standard deviation keys to verify that x=15.8cm and s=3.5cm.

(B). Compute an 80% confidence interval for the population mean of u of rim diameters for such ceramic vessels found at the Wind Mountain archaeological site.

Part 5

2. What do we mean when we say a test is significant? Does this necessarily mean the results are important?

4. All other conditions being equal, does a z or t value of the sample test statistic?

6. Professor Jennings claims that only 35% of the students at Flora College work while attending school. Dean renata thinks that the professor has underestimated the number of students with part-time or full-time jobs. A random sample of 81 students shows that 39 have jobs. Do the data indicate that more than 35% of the students have jobs? (Use a 5% level of significance.)

(A). what is the level of significance? State the null and the hypotheses.

(B). what sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic?

©. Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value.

(D). based on your answers in (A)-(C), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a?

€. Your conclusion in the context of the application.

8. The Nero Match Company sells matchboxes that are supposed to have an average of 40 matches per box, with o=9. A random sample of 94 Nero matchboxes shows the average number of matches per box to be 43.1. Using a 1% level of significance, can you say that the average number of matches per box is more than 40?

(A). what is the level of significance? State the null and the hypotheses.

(B). what sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic?

©. Find (or estimate) the P-value. Sketch the sampling distribution and show the area corresponding to the P-value.

(D). based on your answers in (A)-(C), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a?

€. Your conclusion in the context of the application.

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