Chapter 24, Problem 22E

To determine

To find: The expected cell counts from the two-way table of the referred Exercise 24.9.

Answer to Problem 22E

Solution: The row-wise and column-wise total for a two-way table is shown below:

Opinion	Nature of degree			Total
		Junior College	Bachelor		Graduate
Not scientific at all	44	122	71	237
Purely scientific or a type of scientific	31	62	27	120
Total	75	184	98	357

The table of expected cell counts is obtained as:

Opinion	Expected counts
		Junior College	Bachelor	Graduate
Not scientific at all	49.8	122.2	65.1
Purely scientific or a type of scientific	25.2	61.9	32.9

Explanation of Solution

Calculation:

Form the two-way table, find the row and column totals. The two-way table that contains the row and the column totals is obtained as:

Opinion	Nature of degree			Total
		Junior College	Bachelor		Graduate
Not scientific at all	44	122	71	237
Purely scientific or a type of scientific	31	62	27	120
Total	75	184	98	357

Now, calculate the expected cell counts of each cell. The expected cell count is defined by the formula:

$\text{Expected count}=\frac{\text{row total}\times \text{column total}}{\text{table total}}$

The expected count for Junior college degree adults whose opinion is that astrology is not scientific at all is calculated as:

$\begin{array}{c}\text{Expected count}=\frac{\text{row total}\times \text{column total}}{\text{table total}}\\ =\frac{237\times 75}{357}\\ =\frac{17775}{357}\\ =49.8\end{array}$

The expected count for Bachelor degree adults whose opinion is that astrology is not scientific at all is calculated as:

$\begin{array}{c}\text{Expected count}=\frac{\text{row total}\times \text{column total}}{\text{table total}}\\ =\frac{237\times 184}{357}\\ =\frac{43608}{357}\\ =122.2\end{array}$

Similarly, calculate the expected counts for all cells and it is tabulated as:

Opinion	Expected counts
		Junior College	Bachelor	Graduate
Not scientific at all	49.8	122.2	65.1
Purely scientific or a type of scientific	25.2	61.8	32.9

To determine: The observed counts that differ most from the expected counts.

Solution: The observed counts of Junior college degree and graduate degree differ from the expected counts. The observed counts of Bachelor degree is almost the same as expected counts.

Explanation: A survey is conducted on a simple random sample of adults on their opinion on astrology whether it is purely scientific or type of scientific or not at all scientific. The two-way table that shows the counts for the number of people in the sample for three levels of higher education degrees and their opinion on astrology is provided.

The obtained table of expected counts shows that the observed counts are same as expected counts for Bachelor degree.

But the observed counts differ from expected counts for Junior college degree and graduate degree.

(b)

To determine

To find: The chi-square statistic and the cell that contributes maximum to the chi-square statistic.

Answer to Problem 22E

Solution: The chi-square statistic is obtained as 3.602. The cell that contributes maximum to the chi-square statistic is the cell with Junior college degree with the opinion that astrology is purely scientific or a type of scientific.

Explanation of Solution

Calculation:

The chi-square statistic is the measure of the distance of the observed counts from the expected counts in a two-way table. The formula for the chi-square statistic is defined as:

${\chi }^2={{\displaystyle \sum \frac{\left(\text{observed count}-\text{expected count}\right)}{\text{expected count}}}}^2$

Substitute the obtained observed and expected counts for each cell to determine the chi-square statistic. So, the chi-square statistic is calculated as:

$\begin{array}{c}{\chi }^2={{\displaystyle \sum \frac{\left(\text{observed count}-\text{expected count}\right)}{\text{expected count}}}}^2\\ =\left(\begin{array}{c}\frac{{\left(44-49.79\right)}^2}{49.79}+\frac{{\left(122-122.2\right)}^2}{122.2}+\frac{{\left(71-65.1\right)}^2}{65.1}\\ +\frac{{\left(31-25.2\right)}^2}{25.2}+\frac{{\left(62-61.8\right)}^2}{61.8}+\frac{{\left(27-32.9\right)}^2}{32.9}\end{array}\right)\\ =0.673+0.002+0.534+1.335+0.000+1.058\\ =3.602\end{array}$

In the obtained calculations for chi-square statistic of 3.602, it showed that the cell that corresponds to Junior college degree with the opinion that astrology is purely scientific or a type of scientific contributes maximum value of 1.335 to the chi-square statistic.

(c)

Section 1:

To determine

To find: The degrees of freedom.

Answer to Problem 22E

Solution: The degrees of freedom are 2.

Explanation of Solution

Calculation:

The degrees of freedom for a chi-square test are defined as:

$\text{Degrees of freedom}=\left(r-1\right)\times \left(c-1\right)$

Where, r is the number of rows and c is the number of columns in a two-way table.

In the provided problem, the number of rows is 2 and the number of columns is 3. So, the degrees of freedom are calculated as:

$\begin{array}{c}\text{Degrees of freedom}=\left(r-1\right)\times \left(c-1\right)\\ =\left(2-1\right)\times \left(3-1\right)\\ =1\times 2\\ =2\end{array}$

Section 2:

To test: The significance of the chi-square test by using Table 24.1.

Solution: The survey shows a significant association between the nature of degree and the opinion on astrology at the significance level 0.20 and it does not shows a significant association between the nature of degree and the opinion on astrology at the significance level 0.15.

Explanation:

Calculation:

The null $\left(H_0\right)$ and the alternative hypothesis $\left(H_1\right)$ are:

$H_0$: There is no significant association between the nature of degree and the opinion on astrology.

$H_1$: There is significant association between the nature of degree and the opinion on astrology.

Use Table 24.1 of critical values for chi-square test. The obtained chi-square statistic is 3.602, which is larger than the critical value of 3.22 at the significance level of 0.20 and smaller than the critical value of 3.79 for significance level of 0.15 for two degrees of freedom.

If the significance level 0.20 is used, then the calculated value 3.602 is greater than the critical value 3.22, so it can be said that the result is significant. But if the significance level 0.15 is used then the calculated value 3.602 is less than the critical value 3.79, so it can be said that the result is insignificant.

Conclusion:

At the significance level 0.20, the decision is to reject the null hypothesis, so it can be said that the survey shows there is some association between the nature of degree and the opinion on astrology. At the significance level 0.15, the decision is to accept the null hypothesis, so it can be said that the survey shows there is no association between the nature of degree and the opinion on astrology.