Question

Refer to the file called Hay Fever. In a survey of 16 communities in western Canada the rate of people under 25 who experience hay fever was recorded. Data are given as rate per 1000 people. In that same study, 14 different communities in western Canada were surveyed, and the rates of hay fever for people over 50 years old was also recorded. Again, data recorded as rate per 1000 people.

Hay Fever File:

hay fever rate < 25	hay fever rate > 50
98	95
125	79
90	110
95	115
120	101
125	100
128	97
117	89
92	112
97	114
123	88
122	85
112	110
127	96
93
88

a) Is the sampling method dependent or independent?

b) State the null and alternate hypothesis we would use to test whether the average hay fever rate for the age group over 50 is lower than the average rate for the group of people under 25 years of age.

c) Calculate the test statistic. Show calculation.

d) Run the appropriate test in Minitab and show output. What is the p-value for the test?

e) Assuming $\alpha$ = 0.05, what are your conclusions for these data?

f) From the Minitab output for this test, what does the 95% confidence interval on the mean difference tell you? Does it agree with the test of hypothesis conclusions? Explain.

g) What assumptions are required for the test results to be valid? Have we met those assumptions? Prove it with appropriate plots.

Answer 1

(a)The sampling method is independent. Xi (b)From the given information, for hay fever rate < 25% 109.5, | Σ(x-xi ): 15.4.4 = 16, for hay fever rate > 50% =-= 99.4, n2 -11.6, andn2-14 (b)Null hypothesis H: Alternative hypothesis HI: A> This is a one-tailed test Level ofsignifcance be α-0.05, then the critical value is to0528-1.701, The acceptance regi on is tj S1.701 109.5-99.4 (c)The test statistic tu" . = 2.0571 s2 115.42 11.42 n2 14 (d) P-value =P(t > 2.0571)= 0.025 (e)P-value less than the level of significance α = 0.05, we rej ect the null hypothesis The value of the test statistic tef - 2.0571 greater than the critical value 00528-1.701,it falls in the rejection region, we reject the null hypothesis Therefore, there is enough evi dence to conclude that the average hay fever rate for the age group over 50 is lower than the average rate for the group of people under 25 years of age. (f)100(1-α)% confidence interval for difference of true means is n n2 95% confidence interval for difference of means is 그+그 n2 1542 11.42 (109.5-99.4)-2.04811100.5-99 109.5-99.4) 2.048 16 14 (0.0449,20.1551) Therefore, 95% two-sided confidence interval on the difference in means is 9 SA 20.1551. Yesit agrees with the test of hypothesis conclusions, Since, the confidence interval does not contain zero.