STA 6166 – Exam 1 – Fall 2012 PRINT Name ______

Conduct all tests at a = 0.05 significance level.

Q.1 The probability of randomly selecting the correct response on a multiple choice question with five choices is 0.20 (assuming zero knowledge). Suppose an exam consists of 6 multiple choice questions, each with five choices.

p.1.a. How many correct responses would you expect a student to pick by randomly selecting answers?

p.1.b. What is the probability a student gets none of the questions correct?

Q.2 A bridge holds up to 25 cars. It is known the weight of individual cars is normally distributed with m = 2100 lbs. and s = 500 lbs.

p.2.a. What is the sampling distribution of the sample mean weight for n=25 cars?

p.2.b. If the maximum load for which the bridge is designed is 80,000 lbs., what is the probability a full load of cars (n=25) will exceed the design limits?

Q.3. Find the mean, median, standard deviation for the following data.

16 10 18 15 11 9 5

Q.4 A veterinarian is interested in estimating the average amount she spends monthly on vaccines. She records the dollar amount for 25 randomly chosen months and finds the mean to be $370 and the sample standard to be $25. Give a point estimate and a 95% confidence interval for the population mean monthly amount spent on vaccines by this veterinarian.

Q.5. A home gardener (a statistician) plants two varieties of tomato plants. Variety B is advertised to produce higher yields. During the harvest season he records the yields of the individual plants. Do the data show significant evidence to support the claim that variety B produces a higher median yield? Use the Wilcoxon Rank-Sum Test with a = 0.05.

Variety A 40.3 (1) 51.3 (4) 54.4 (6) 46.8 (2) 49.2 (3)

Variety B 55.4 (9) 55.2 (8) 56.0 (10) 53.4 (5) 54.5 (7)

Rank Sum for Variety A= 1+4+6+2+3 = 16

Rank Sum for Variety B 9+8+10+5+7 = 39

Conclude Variety B higher if Rank Sum for A ≤ 19 (1-sided, a = 0.05)

Q.6. You want to estimate the difference between two means within + 20 of the true population mean difference with 95% confidence (that is, E=20). Assuming the common standard deviation is 15, how many observations should you select from each population, based on independent samples?

Q.7. A study measured a physical characteristic of identical twins. The purpose of the study to was to determine if there was a significant difference in the means for the younger twin versus the older twin. A summary of the data is (Difference = Younger - Older for each pair).

Twin N Mean Std. Dev.

Younger 16 1.76 0.24

Older 16 1.56 0.30

Difference 16 0.20 0.24

Is there significant evidence that younger twins differ on average from older twins? H0: mD= 0 vs HA: mD≠ 0

Note: This is a Paired difference test (Twins=pairs)

p.7.a. Test Statistic:

p.7.b. Reject H0 if the test statistic falls in the range(s) |tobs| ≥ t.025,15 = 2.132

p.7.c. The P-value is larger than or smaller than 0.05

Q.8. A diagnostic test has 95% sensitivity (the probability a person with the condition tests positive = 0.95) and 95% specificity (the probability a person without the condition tests negative = 0.95). In a population of people given the test, 1% of the people have the condition (probability a person has the condition = 0.01).

p.8.a. What proportion of the people will test positive?

p.8.b. Given a person has tested positive, what is the probability he/she has the condition?

Q.9. You will be conducting a study to compare a measure of fitness for two types of exercise programs. You will use a significance level = 5%. If the two means differ by more than 5 units in either direction, you want to have a probability of at least 90% of rejecting Ho. Assuming a common s = 4.0, what is the required sample size for each program, based on independent samples? Assume for now, the sample and population means are higher for program 1