Question

For each scenario below, do the following:

1. (1) Indicate which procedure you would use: (i) two-independent sample t-test, (ii) paired t- test, (iii) chi-square goodness-of-fit test, (iv) chi-square test of independence, (v) t-test of a regression coefficient, (vi) an F −test for multiple regression, (vii) a one-way ANOVA, or (viii) a two-way ANOVA.
2. (2) Based on your answer in (1), state the appropriate null and alternative hypotheses.
3. (3) For all procedures except (i), state the degrees of freedom(s) for the corresponding test statis- tic.

(a) Based on historical data, the percentages of people who apply for the Silver, Gold, and Platinum membership at an aquatic and fitness center are 55%, 32%, and 13%, respectively. In the past 6 months, there have been 102 applications for the Silver level, 58 applications for the Gold level, and 40 applications for the Platinum level. Is there evidence that the percentages observed in the past 6 weeks is different that what has been observed historically?

(b) A home gardener is studying the effect of 10 different fertilizers on the growth of Kentucky bluegrass (a particular variety of grass seed). He plants 12 seeds in 10 different pots,and adds the same amount of fertilizer to each pot. After two weeks he measures the heights of the 120 grasses in millimeters.

(c) A basketball player performed an experiment to see if her favorite shoes and the time of day might affect her free throw percentage. In particular, she wants to see if there is an interaction effect between wearing her favorite shoes (with or without) and the time of day (early morning or at night). For each treatment combination, she shot 50 baskets on 4 different occasions, recording the number of shots made each time. She randomized the treatment conditions by drawing numbers out of a hat.

(d) Does premium octane gasoline improve gas mileage? Suppose an experiment is conducted to test the claim that cars running on premium gas get better gas mileage than the same car running on regular gas. In the experiment, 10 cars are used (all cars are the same make and model). For each car, a coin is tossed. If heads comes up, then the car: first drives on 12 gallons of regular gas, the mileage driven is measured, then the car drives on 12 gallons of premium gas, after which the mileage driven is measured. If the coin comes up tails, the same procedure is used, except premium gas is used first. The experiment is blinded in that the drivers are not told the type of gas in the car.

Answer 1

a)
chi-square goodness-of-fit test

Ho : the data fits the given distribution
Ha: the data does not fit the given distribution

df =k-1 = 2

b)
ANOVA one way

Ho: µ1 = µ2= ...= µ10

Ha: at least one mean is different

df1 = k-1 = 10-1 = 9
df2 = N-k = 12*10 - 10 = 110

c)
two way anova

Ho: There is no difference in favorite shoes

Ha : There is difference in favorite shoes

Ho: There is no difference in time of day

Ha : There is difference in time of day

Ho : there is no interaction

Ha: there is an interaction

df = 1 for both main effects and interaction

df Error = 50*4 - 1 - 3 = 196

d)
paired t-test

Ho : mu_d = 0

Ha : mu_d > 0
df = n-1 = 10-1 = 9