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

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

Test the goodness-of-fit at the $\alpha=0.05$ significance level.

A group of children are asked to choose their favorite toy from a ball, a doll, and a puzzle. Out of 90 responses, 35 chose the ball, 25 chose the doll, and 30 chose the puzzle. Perform a chi-squared goodness of fit test at the $5\%$ significance level to determine whether the preferences for the three toys are equally likely.

Over 210 births, the number of babies born on each day of the week is recorded as: Monday 31, Tuesday 35, Wednesday 30, Thursday 38, Friday 33, Saturday 28, and Sunday 15.

Test if births are equally likely on each day at the 5% significance level.

A random sample of 100 data points records the last digit of each measurement. The observed counts for the digits 0 to 9 are 12, 8, 11, 9, 10, 7, 14, 6, 11, and 12, respectively.

Test, at the 5% significance level, whether the last digits follow a uniform distribution.

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

In a survey of 150 people, the frequencies of hair colors were found to be: Black 50, Brown 60, Blonde 30, and Red 10. The known proportions for these hair colors in the population are 0.35, 0.30, 0.25, and 0.10 respectively.

Test, at the 5% significance level, whether the hair color distribution in the survey is consistent with the population proportions.

A fair six-sided die is rolled 60 times. The observed frequencies for faces 1 through 6 are recorded as: 8, 12, 9, 11, 10, and 10.

Test the hypothesis that the die is fair at the 5% significance level. In your response, calculate the chi-squared statistic and the $p$-value, and state your conclusion.

A quality‐control inspector records the number of defects per item in a sample of 150 items. The counts are summarized in the following table:

Assuming that the number of defects follows a Poisson distribution with $\lambda = 1$, perform a chi-squared goodness-of-fit test at the 5% significance level to determine if this model is appropriate for the data.

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

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