Math 217N - Hanover College
Math 217 Name: __________________
12-2-05
Exam 3, Chapters 5 & 6
1. (5 pts) A couple decides to continue to have children until their first girl is born; X is the total number of children the couple has. Is X binomial? ______
Explain:
2. (10 pts) The administration of your college claims that 67% of all students at your college support a crackdown on underage drinking. To test this claim, you survey an SRS of 60 students (using random mailbox numbers) and find that 32 out of 60 students surveyed support a crackdown on underage drinking.
a) What is the sample proportion who support a crackdown on underage drinking? _____
b) If in fact the proportion of all students who support a crackdown is 67%, what is the probability that the proportion in an SRS of 60 students is as small or smaller than the result of your survey (32 of 60)? __________
Show your work:
3. (5 pts) A factory employs several thousand workers, of whom 30% are Hispanic. If the 15 members of the union executive committee were chosen from the workers at random, the number of Hispanics on the committee would have a binomial distribution. What is the probability that 3 or fewer members a randomly-chosen executive committee are Hispanic? ___________
Show your work:
4. (10 pts) The idea of insurance is that we all face risks that are unlikely but carry high cost. So we form a group to share the risk; we all pay a small amount, and the insurance company gives a large amount to those few of us who suffer disasters.
An insurance company looks at the records for millions of homeowners and sees that the mean loss from fire in a year is $250 per house and that the standard deviation of the loss is $1000. The distribution of losses is extremely right-skewed since most people have no loss but a few have large losses. The company plans to sell fire insurance for $250 per year plus enough to cover its costs and profit, say $300 per policy.
(a) If the company sells 100 policies, what is the approximate probability that the average loss in a year will be greater than $275? ____________
Show your work:
(b) If the company sells 40,000 policies, what is the approximate probability that the average loss in a year will be greater than $275? ____________
Show your work:
5. (10 pts) Statisticians prefer large samples. Describe briefly the effect of increasing the sample size (or the number of subjects in an experiment) on each of the following:
a) The width of a 95% confidence interval.
b) The P-value of a significance test, when the null hypothesis is false.
c) The variability of the sampling distribution of [pic].
d) The mean of the sampling distribution of [pic].
e) The shape of the sampling distribution of [pic], even if the original variable X is highly skewed.
6. (5 pts) You are planning a sample survey of households Jefferson County, Indiana to determine the average monthly household food budget for all such households. A preliminary study suggests that the standard deviation in monthly household food budgets in Jefferson County is about $350. If you want to estimate the average monthly household food budget with a 95% confidence interval and a margin of error no more than $50, what is the minimum sample size you will need? _________
Show your work:
7. (5 pts) You want to rent an apartment in Dallas. The mean monthly rent for a random sample of 20 unfurnished one-bedroom apartments advertised in a Dallas newpaper is $723. Assume that the standard deviation of the population is $90 and that the population is normally distributed. Find a 90% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments available for rent in Dallas. ______________________________
Show your work:
8. (5 pts) The following situation calls for a significance test for a population mean µ. State the null hypothesis [pic]and the alternative hypothesis [pic].
Situation: Experiments on learning in animals sometimes measure how long it takes a mouse to find its way through a maze. The mean time is 18 seconds for one particular maze. A researcher thinks that a loud noise will cause the mice to complete the maze faster. She measures how long each of 10 mice takes with a loud noise as stimulus.
[pic](in math symbols): _______________[pic] (in math symbols): _______________
9. (10 pts) To determine whether the mean nicotine content of a brand of cigarettes is greater than the advertised value of 1.4 milligrams, a health advocacy group conducts a test of significance. The calculated value of the test statistic is z = 1.09.
(a) What is the P-value? _______________
(b) What conclusion should be reached about the nicotine content of this brand of cigarettes?
10. (5 pts) Your pharmaceutical company has discovered a new drug, Reducalot, for weight loss. You pay for 100 different studies to compare Reducalot with a placebo pill. Three of the studies show that Reducalot is significantly more effective (P < .05) than a placebo. Is it correct to conclude (and publicize) that Reduca is an effective drug for inducing weight loss? ___________ Explain.
11. (5 pts) True or false? If the P-value from a test of significance is more than .05, then we do not have strong evidence in support of the alternative hypothesis. ____________
Explain:
12. (5 pts) True or false? If the P-value from a test of significance is .6355, then the null hypothesis is true. ____________
Explain:
5.1 Formulas: Binomial Distribution, Sample Count, Sample Proportion
Note: The formulas below require that all random sampling is done from a much larger population (population size at least 10 times larger than sample size).
1. Binomial mean and standard deviation. If a count X has the binomial distribution B(n, p):
• the mean of X is[pic]
• the standard deviation of X is[pic]
2. Sample count is binomial. The binomial distribution B(n, p) is a good approximation to the sampling distribution of the count of successes (X) in an SRS of size n from a population containing proportion p of successes. Hence, [pic] and [pic].
3. Small n method for a count X. When n is reasonably small, probabilities for a sample count X can be found in Table C or by using binompdf(n, p) on your calculator.
4. Large n method for a count X. When n is large enough that both np and [pic]are at least 10, the normal distribution N( np,[pic] ) is a good approximation to the sampling distribution of the count of successes (X). Probabilities can be found in
Table A.
5. Sample proportion mean and standard deviation. The sample proportion of successes ([pic]) in an SRS of size n from a population containing proportion p of successes has mean and standard deviation as follows:
• [pic]
• [pic]
6. Probabilities for sample proportion. Note that[pic]does not follow a binomial distribution. Probabilities for [pic] can be found in two ways. The first way has the advantage of working for all n, not just for large n.
i. Convert a question about [pic]to a corresponding question about X (sample count) and then use the sample count procedures shown above.
ii. When n is large enough that both np and [pic]are at least 10, the sampling distribution of [pic]is approximated by the normal distribution N( p,[pic] )
(find probabilities in Table A).
5.2 Formulas: Distribution of the Sample Mean
The sample mean [pic] of an SRS of size n drawn from a large population with mean [pic]and standard deviation [pic]has a sampling distribution with the following properties:
• mean [pic] (sample mean is an unbiased estimator of the population mean)
• standard deviation [pic] (to cut the variability of the sample mean in half, quadruple the sample size).
• If n is large then [pic] is approximately normally distributed (this is the Central Limit Theorem).
• If the original variable X is normally distributed, then [pic] is normally distributed.
6.1 Formulas: Confidence Interval for Population Mean (σ known)
• A level C confidence interval for the mean μ of a normal population with known standard deviation σ, based on an SRS of size n, is given by [pic]. If the population is not normally distributed then the sample size should be large (at least 40). z* is obtained from the bottom row in Table D:
z* |0.674 |0.841 |1.036 |1.282 |1.645 |1.960 |2.054 |2.326 |2.576 |2.807 |3.091 |3.291 | |C |50% |60% |70% |80% |90% |95% |96% |98% |99% |99.5% |99.8% |99.9% | |
• The minimum sample size required to obtain a confidence interval of specified margin of error m for a normal mean μ is given by [pic]where z* is obtained from the bottom row in Table D according to the desired level of confidence.
6.2 Formulas: Z Test for a Population Mean
a. Left-tail Z Test for a Population Mean:
1. State the null hypothesis [pic]: [pic]. ([pic] is a specific number, the cut-off value between the two competing claims.)
2. State the alternative hypothesis [pic]. ([pic] is that same specific number as in the null hypothesis. The alternative hypothesis is the researcher’s claim. The researcher hopes to provide convincing evidence that we can reject the null hypothesis and accept the alternative hypothesis.)
3. Based on an SRS of size n from the population, calculate the sample mean [pic] and the test statistic [pic]. (This test requires σ to be known. Also, if the population is not normally distributed then the sample size should be at least 40.)
4. Find the P-value, [pic][left-tail area for z], in Table A.
5. If P is very small (less than .05? less than .01?) then we have strong enough evidence to reject the null hypothesis and accept the alternative hypothesis.
b. In a right-tail Z test, the alternative hypothesis has the form [pic] and the
P-value is [pic], the right-tail area for z.
c. In a two-tail Z test, the alternative hypothesis has the form [pic] and the
P-value is [pic], the two-tail area for z.
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