Hypothesis test 4?

[TerryC7569]

In the year 2000, the state of Indiana began a $40-million renovation of its state fairgrounds, which included the building of a miniature golf course and a state-of-the-art livestock building. Now, Indiana officials are interested in learning what sorts of people are visiting the new attractions. In a survey done at this year's state fair, it was found that, among a random sample of 71 couples at the fair with their children, 43 had visited the new miniature golf course, and among an independently chosen, random sample of 66couples at the fair on a date (without children), 33 had visited the miniature golf course. Based on these samples, can we conclude, at the 0.1 level of significance, that the proportion p1 of all couples attending the fair with their children who visited the miniature golf course is different from the proportion p2 of all couples attending the fair on a date who visited the miniature golf course? Perform a two-tailed test.1. the null hypothesis H. 2. the alternative hypothesis H:1. 3. the type of test statistic. 4. the value of the test statistic. 5. the p-value:6. can we conclude that the proportion of couples visiting the miniature golf course is different between the two groups? Yes or No

[Merlyn]

Large Sample Hypothesis Test for the Difference in Proportions Let X be the number of success in nx independent and identically distributed Bernoulli trials, i.e., X ~ Binomial(nx, px) Let Y be the number of success in ny independent and identically distributed Bernoulli trials, i.e., Y ~ Binomial(ny, py) LetpxHat = X / nx pyHat = Y / ny pHat = (X + Y) / (ny + ny) Assuming that (nx + ny)*pHat > 10 and (nx + ny)*(1-pHat) > 10 (some will say the necessary condition here is > 5, I prefer this more conservative assumption so that the approximations in the tail of the distribution are more accurate), then to test the null hypothesis H0: px - py = Δ or H0: px - py = Δ or H0: px - py = Δ Find the test statistic z = ((pxHat - pyHat) - Δ ) / (sqrt(pHat * (1 - pHat) * (1/nx + 1/ny)) The p-value of the test is the area under the normal curve that is in agreement with the alternate hypothesis. H1: px - py > Δ; p-value is the area to the right of z H1: px - py < Δ; p-value is the area to the left of z H1: px - py â‰* Δ; p-value is the area in the tails greater than |z| If the p-value is less than or equal to the significance level α, i.e., p-value ≤ α, then we reject the null hypothesis and conclude the alternate hypothesis is true. If the p-value is greater than the significance level, i.e., p-value > α, then we fail to reject the null hypothesis and conclude that the null is plausible. Note that we can conclude the alternate is true, but we cannot conclude the null is true, only that it is plausible.The hypothesis test in this question is:H0: px - py = 0 vs. H1: px - py â‰* 0The test statistic is:z = (( 0.6056338 - 0.5 ) - 0 ) / ( √ ( 0.5547445 * (1 - 0.5547445 ) * ( 1 / 71 + 1 / 66 ) z = 1.243062The p-value = P( Z > |z| )= P( Z < -1.243062 ) + P( Z > 1.243062 )= 2 * P( Z < -1.243062 )= 0.2138452Since the p-value is greater than the significance level of 0.1 we fail to reject the null hypothesis and conclude px - py = 0 is plausible.