a social service organization reports that th...
a social service organization reports that the level of educational attainment of mothers receiving food stamps is uniformly distributed. to test this claim, you randomly select 100 mothers who currently receive food stamps and record the educational attainment of each. the results are shown in the table on the right. at $alpha = 0.025$, can you reject the claim that the distribution is uniform? complete parts (a) through (d) below.\nresponse frequency, $f$\nnot a high school graduate 33\nhigh school graduate 40\ncollege (1 year or more) 27\n(b) determine the critical value, $chi_{0}^{2}$, and the rejection region.\n$chi_{0}^{2}=7.378$ (round to three decimal places as needed.)\nchoose the correct rejection region below.\na. $chi^{2}geqchi_{0}^{2}$\nb. $chi^{2}leqchi_{0}^{2}$\nc. $chi^{2}>chi_{0}^{2}$\nd. $chi^{2}<chi_{0}^{2}$\n(c) calculate the test statistic.\n$chi^{2}=square$ (round to three decimal places as needed.)
Answer
# Explanation:
## Step1: Calculate expected frequencies
Since there are 3 categories and a total of $n = 100$ observations, if the distribution is uniform, the expected frequency for each category is $E=\frac{100}{3}\approx33.333$.
## Step2: Calculate the $\chi^{2}$ - test statistic formula
The formula for the $\chi^{2}$ - test statistic is $\chi^{2}=\sum\frac{(O - E)^{2}}{E}$, where $O$ is the observed frequency.
For the first category (Not a high - school graduate): $O_1 = 33$, $E_1=33.333$, $(O_1 - E_1)^{2}=(33 - 33.333)^{2}=(- 0.333)^{2}=0.111$, and $\frac{(O_1 - E_1)^{2}}{E_1}=\frac{0.111}{33.333}\approx0.00333$.
For the second category (High - school graduate): $O_2 = 40$, $E_2 = 33.333$, $(O_2 - E_2)^{2}=(40 - 33.333)^{2}=(6.667)^{2}=44.44889$, and $\frac{(O_2 - E_2)^{2}}{E_2}=\frac{44.44889}{33.333}\approx1.333$.
For the third category (College (1 year or more)): $O_3 = 27$, $E_3 = 33.333$, $(O_3 - E_3)^{2}=(27 - 33.333)^{2}=(-6.333)^{2}=40.10689$, and $\frac{(O_3 - E_3)^{2}}{E_3}=\frac{40.10689}{33.333}\approx1.203$.
## Step3: Sum up the values
$\chi^{2}=0.00333 + 1.333+1.203=2.539$.
# Answer:
$2.539$