Question Type 8: Calculating the chi-squared value and/or the p-value to conclude the chi-squared goodness of fit test Exercises

A coin is tossed 100 times, yielding 55 heads and 45 tails. Test whether the coin is fair using a chi-squared test at $\alpha=0.05$.

Children are asked their favorite toy among ball, doll, and puzzle. Out of 90 responses, 35 chose ball, 25 doll, and 30 puzzle. Using a chi-squared test at $\alpha=0.05$, test if preferences are equally likely.

Over 210 births, the number of babies born on each day of the week is recorded as: Mon 31, Tue 35, Wed 30, Thu 38, Fri 33, Sat 28, Sun 15. Test if births are equally likely on each day at the 5% significance level.

A fair six-sided die is rolled 60 times. The observed frequencies for faces 1 through 6 are: 8, 12, 9, 11, 10, 10. Test whether the die is fair by calculating the chi-squared statistic, the p-value, and conclude at the 5% significance level.

A fruit vendor claims that sales proportions of apples, bananas, cherries, and pears are 0.30, 0.25, 0.25, and 0.20. A random sample of 200 sales records shows 62 apples, 47 bananas, 50 cherries, and 41 pears. Test the claim at the 5% level.

A random sample of 100 data points records the last digit of each measurement: counts for digits 0–9 are 12, 8, 11, 9, 10, 7, 14, 6, 11, 12. Test uniformity of last digits at the 5% level.

A researcher obtains $\chi^2=2.71$ with $df=4$. At $\alpha=0.05$, decide whether to reject $H_0$.

In a survey of 150 people, hair color counts are: Black 50, Brown 60, Blonde 30, Red 10. The known population proportions are 0.35, 0.30, 0.25, and 0.10 respectively. Test at 5%.

You sample 100 M&M chocolates and record the following observed counts: Blue 23, Brown 14, Green 15, Orange 18, Red 12, Yellow 18. The manufacturer claims the color proportions are 0.24, 0.13, 0.16, 0.20, 0.13, 0.14 respectively. Perform a chi-squared goodness-of-fit test at the 5% level.

A test statistic yields $\chi^2=7.81$ with $df=3$. Determine the p-value and state the conclusion at the 5% level.

A manufacturing process yields defects per item following a Poisson distribution with mean $\lambda=2$. In a sample of 200 items, counts of defects are recorded as: 0 defects: 40 items; 1 defect: 80; 2 defects: 50; 3 defects: 20; 4 defects: 5; 5 or more defects: 5. Test the fit at $\alpha=0.05$.

A quality‐control inspector records the number of defects per item in a sample of 150 items. The counts are: 0 defects: 60; 1 defect: 55; 2 defects: 25; 3 or more defects: 10. Assume defects follow Poisson($\lambda=1$). Test at $\alpha=0.05$.