A recent Health of Boston report suggests that 14% of female residents suffer from asthma as opposed to 6% of males. Suppose 250 females and 200 males responded to the study. Which of the following is the correct value of the test statistics?
A) −1.76
B) 2.76
C) −2.76
D) 1.76

Question

Quizplus · Accepted Answer

We can use a chi-square test to determine if there is a statistically significant difference between the proportion of females and males with asthma.

First, we need to set up a contingency table:

Asthma   No Asthma   Total
Females     35       215        250
Males       12       188        200
Total       47       403        450

We then calculate the expected values for each cell in the table, assuming the null hypothesis that there is no difference between the two groups:

Asthma   No Asthma   
Females     250*47/450      250*403/450
            = 26.11          = 223.89
Males       200*47/450      200*403/450
            = 20.89          = 179.11

We can then calculate the chi-square statistic:

χ² = (35 - 26.11)²/26.11 + (215 - 223.89)²/223.89 + (12 - 20.89)²/20.89 + (188 - 179.11)²/179.11
   = 6.86

Using a chi-square distribution table with 1 degree of freedom (since we are comparing two groups), we find that the critical value at the 0.05 level of significance is 3.84. Since our calculated chi-square value (6.86) is greater than the critical value, we can reject the null hypothesis and conclude that there is a statistically significant difference between the proportion of females and males with asthma.

To find the test statistic, we can use the formula:

test statistic = (observed - expected) / sqrt(expected)

For the cell with females and asthma: 
test statistic = (35 - 26.11) / sqrt(26.11) = 2.74

For the other cells: 
test statistic for females and no asthma = (215 - 223.89) / sqrt(223.89) = -1.28 
test statistic for males and asthma = (12 - 20.89) / sqrt(20.89) = -2.97 
test statistic for males and no asthma = (188 - 179.11) / sqrt(179.11) = 1.89

To get the overall test statistic, we can take the square root of the sum of the squared individual test statistics:

test statistic = sqrt(2.74² + (-1.28)² + (-2.97)² + 1.89²) = 2.76

Therefore, the correct value for the test statistic is B) 2.76.

A Recent Health of Boston Report Suggests That 14% of Female