A number of grade school students were asked about their perceived math ability and the results are in the table below.
Below Average = BA
Average = A
Above Average = AV
BA A AV
Girls 61 54 21
Boys 43 61 42
Test at the 0.01 level if there is a difference between girls and boys with respect to their perceived math ability.
Use the Chi-square (X^2) method.
X^2 = ∑ (O-E)^2/E, where O = observed frequency and E = expected frequency.
∑ = sum of all the cells.
E = (column total * row total)/grand total
df = n - 1, where n = number of cells
Look up value in X^2 table in the back of your textbook.
To test if there is a difference between girls and boys with respect to their perceived math ability, we can use the Chi-square test of independence. This test assesses if there is a relationship between two categorical variables, in this case, gender (girls and boys) and perceived math ability (below average, average, above average).
Step 1: Set up the null and alternative hypotheses:
- Null Hypothesis (H0): There is no difference between girls and boys with respect to their perceived math ability.
- Alternative Hypothesis (H1): There is a difference between girls and boys with respect to their perceived math ability.
Step 2: Calculate the expected frequencies:
We need to calculate the expected frequencies for each cell in the table. The expected frequency is the value we would expect if there was no relationship between the variables. To calculate the expected frequency for each cell, we use the formula:
Expected Frequency = (row total * column total) / grand total
Step 3: Perform the Chi-square test:
Now, we can perform the Chi-square test using the observed and expected frequencies. The test statistic is calculated as:
Chi-square statistic = Σ ( (O - E)^2 / E )
Where Σ represents the sum of the calculations for each cell, O is the observed frequency, and E is the expected frequency.
Step 4: Determine the critical value and p-value:
In this step, we compare the calculated chi-square statistic to the critical value from the chi-square distribution table with (r-1) * (c-1) degrees of freedom, where r is the number of rows and c is the number of columns in the table. We also determine the p-value associated with the test statistic.
Step 5: Make a decision and interpret the results:
If the calculated chi-square statistic is greater than the critical value or if the p-value is less than the chosen significance level (0.01 in this case), we reject the null hypothesis. This would indicate that there is evidence of a difference between girls and boys with respect to their perceived math ability.
Please provide the observed frequencies for each cell so that I can help you perform the calculations and interpret the results.