Question Type 6: Performing the chi-squared test for data categorization such that the expected frequency is at least 5 Exercises

A six-sided die is rolled $180$ times with the following outcomes:

Test at the $1\%$ significance level whether the die is fair.

A psychologist classifies 140 subjects by coping style (Problem-focused, Emotion-focused, Avoidant) and stress level (Low, Medium, High). The observed frequencies are shown in the following table:

Perform a $\chi^2$ test of independence at the $5\%$ level of significance to determine whether coping style is independent of stress level.

In a genetics experiment, 120 plants are classified by flower color: purple, pink, or white. The expected Mendelian ratio is $9:3:4$. Observed counts are 54 purple, 20 pink, and 46 white.

Test at the $5\%$ significance level whether the data fit this ratio.

A group of 120 students are asked to choose one of four teaching methods. The number of students who chose each method is shown in the table below.

$\begin{array}{|c|c|c|c|c|} \hline \text{Method} & \text{I} & \text{II} & \text{III} & \text{IV} \\ \hline \text{Number of students} & 28 & 34 & 26 & 32 \\ \hline \end{array}$

Perform a $\chi^2$ goodness of fit test, at the 5% significance level, to determine whether students have a preference for any of the teaching methods.

A manufacturer claims that $40\%$ of its light bulbs last at least $800$ hours, $35\%$ last between $500$ and $799$ hours, and $25\%$ last less than $500$ hours. In a test of $200$ bulbs, $68$ lasted $\geq 800$ h, $70$ lasted $500\text{--}799$ h, and $62$ lasted $< 500$ h.

Test the claim at the $5\%$ level.

In a study of 200 patients, blood type (A, B, AB, O) and the presence of a certain antibody (positive or negative) were recorded. The results are shown in the following table.

A $\chi^2$ test for independence is performed at the $5\%$ significance level.

(a) State the null hypothesis, $H_0$, for this test.

(b) Show that the expected frequency for a patient having blood type O and testing positive for the antibody is 15.

(c) Calculate the $\chi^2$ test statistic for this data.

(d) State the number of degrees of freedom for this test.

(e) The critical value for this test is 7.815. State the conclusion of the test, giving a reason for your answer.

A retailer claims that 20% of customers choose brand A, 30% brand B, 25% brand C, and 25% brand D. In a random sample of 200 purchases, the counts are 38, 64, 50, and 48 respectively. Test the claim at the 5% significance level.

A snack company claims that customers choose its three chip flavors—salted, barbecue, and sour cream & onion—with equal probability. In a survey of $150$ customers, the observed counts are $42$ salted, $58$ barbecue, and $50$ sour cream & onion. Test at the $5\%$ significance level whether the flavor choices are equally likely.