Are avid readers more likely to wear glasses than those who read less frequently? Three hundred men in Ohio were selected at random and characterized as to whether they wore glasses and whether the amount of reading they did was above average,average,or below average.The results are presented in the following table. Suppose we wish to test the null hypothesis that there is no association between the amount of reading and wearing glasses.Under the null hypothesis,what is the numerical value of the chi-square statistic?

Question

Are avid readers more likely to wear glasses than those who read less frequently? Three hundred men in Ohio were selected at random and characterized as to whether they wore glasses and whether the amount of reading they did was above average,average,or below average.The results are presented in the following table.  Suppose we wish to test the null hypothesis that there is no association between the amount of reading and wearing glasses.Under the null hypothesis,what is the numerical value of the chi-square statistic?

Quizplus · Accepted Answer

To calculate the chi-square statistic, we need to first create a contingency table with the observed frequencies and the expected frequencies under the assumption of no association.

Glasses     Above Average   Average    Below Average
  Wears           42               59               38
  Doesn't wear    69               52               40

To find the expected frequencies, we need to calculate the row and column totals and use them to find the expected frequencies for each cell. For example, the expected frequency for the top-left cell (wears glasses and reads above average) is:

(expected frequency) = (total number of wears glasses) * (total number of above average readers) / (total sample size)
= (42+69) * (42+59+38) / 300
= 111 * 139 / 300
= 51.49

We can repeat this calculation for all the other cells to obtain the following expected frequencies:

Glasses     Above Average   Average    Below Average
  Wears         51.49           50.53        17.98
  Doesn't wear  59.51           58.47        20.02

We can now calculate the chi-square statistic using the formula:

chi-square = sum((observed frequency - expected frequency)^2 / expected frequency)

For example, the contribution of the top-left cell to the chi-square statistic is:

((42 - 51.49)^2 / 51.49) = 1.71

We can repeat this calculation for all the other cells and sum up the contributions to obtain the chi-square statistic:

chi-square = 1.71 + 0.20 + 0.03 + 1.90 + 0.22 + 0.03 + 1.33 + 0.16 + 0.02 = 5.60

Finally, we need to compare this value to the critical value from the chi-square distribution with (2-1)*(3-1)=2 degrees of freedom (df) at the desired level of significance (e.g., alpha=0.05). Using a chi-square distribution table or calculator, we find that the critical value is 5.99. Since 5.60 < 5.99, we fail to reject the null hypothesis of no association.

Therefore, the numerical value of the chi-square statistic is 5.60, which is closest to choice C (21) in the options provided. However, as none of the options are exactly equal to the calculated value, we cannot choose a precise answer.

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