Question

Define the sets \(A_{1}=\\{x:-\infty

Short answer

Whether \(H_{0}\) would be accepted or rejected at the 5% level of significance is determined by comparing the p-value calculated in the chi-square test with the level of significance. This must be done by following the prescribed steps in R.

Step-by-step solution

Calculate the Expected Frequencies

First, calculate the expected frequencies of the sets \(A_{i}\) based on the given probabilities \(p_{i0}\). This can be done using the integral equation given and the R function pnorm(). For example, the expected frequency for set \(A_3\) is \(p_{30}\), which can be computed in R as pnorm(2,3,2)-pnorm(1,3,2). Repeat this for all sets \(A_{i}\).

Perform the Chi-Square Test

Perform a chi-square test to compare the expected frequencies computed in Step 1 with the observed frequencies given in the question. This can be done using the R function chisq.test(). The input to this function should be a table of observed and expected frequencies.

Determine the Level of Significance

The chi-square test will provide a p-value. If the p-value is less than the level of significance (in this case, 5%), then we reject \(H_{0}\), and if the p-value is greater than the level of significance, we accept \(H_{0}\).

Interpret the Result

Depending on whether \(H_{0}\) was accepted or rejected, interpret the result in context of the problem. If the calculated chi-square statistic is less than the critical value of chi-square, then we cannot reject the hypothesis at the given level of significance.