7. A random sample of n1 = 16 communities in western Kansas gave the following rates of hay fever per 1000 population for people under 25 years of age.
121 115 124 99 134 121 110 116 113 96 116 116 135 96 96 116
A random sample of n2 = 14 communities in western Kansas gave the following rates of hay fever per 1000 population for people over 50 years old.
113 86 106 102 113 94 94 108 103 99 78 105 88 100
Assume that the hay fever rate in each age group has an approximately normal distribution. Using the method outlined in Brase and Brase, do the data indicate that the age group over 50 has a lower rate of hay fever? Use Do you reject or fail to reject the null hypothesis? Are the data statistically significant at the level of significance?
To determine whether the age group over 50 has a lower rate of hay fever compared to the under 25 age group, we can perform a two-sample t-test. The steps involved are as follows:
Step 1: State the hypotheses.
The null hypothesis (H0) assumes that the means of the two groups are equal, while the alternative hypothesis (Ha) assumes that the mean of the over 50 age group is lower.
H0: μ1 = μ2 (the mean hay fever rate in the under 25 and over 50 age groups are equal)
Ha: μ1 > μ2 (the mean hay fever rate in the over 50 age group is lower)
Step 2: Set the level of significance (α).
The level of significance determines the probability of making a Type I error. In this case, we can choose a level of significance, such as α = 0.05, which is commonly used in hypothesis testing.
Step 3: Compute the test statistic.
The two-sample t-test formula is given by:
t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))
Where:
- x1 and x2 are the sample means of the under 25 and over 50 age groups, respectively.
- s1 and s2 are the sample standard deviations of the under 25 and over 50 age groups, respectively.
- n1 and n2 are the sample sizes of the under 25 and over 50 age groups, respectively.
Step 4: Determine the critical value(s) and rejection region.
Since the alternative hypothesis assumes that the mean hay fever rate in the over 50 age group is lower, we are dealing with a one-tailed test. We need to find the critical value for a one-tailed t-test with the given level of significance and degrees of freedom (df) equal to (n1 - 1) + (n2 - 1).
The rejection region is the left tail of the t-distribution.
Step 5: Compare the test statistic with the critical value.
If the test statistic falls in the rejection region (i.e., less than the critical value), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
To perform these calculations, let's input the given data into statistical software or a calculator with built-in statistical functions to obtain the test statistic and p-value. Without this information, it is not possible to determine the outcome of the hypothesis test or conclude whether the data is statistically significant at the given level of significance.