a. A researcher is interested in determining if

Introduction:
As it is of interest to test the presence of any relationship between handedness and penmanship, the null and alternative hypotheses would be:
${\displaystyle H_0}$: There is no relationship between handedness and penmanship, that is, handedness and penmanship are independent.
${\displaystyle H_1}$: There is a relationship between handedness and penmanship, that is, handedness and penmanship are not independent but are associated.
a. Clearly, there are two categorical variables- handedness and penmanship, with handedness having two levels and penmanship having three levels. In order to test whether there is any relationship between handedness and penmanship, the researcher would use a test designed to identify the presence of a relationship, or the absence of one.
An ANOVA F-test or analysis of variance F-test is suitable for comparing the means of more than two independent groups.
The two sample z-test and two sample t-test are suitable for comparing the means of two independent groups, with the z-test being used under the assumption of at least approximate normality when the population standard deviations of the two groups are known, and the t-test being used under the assumption of at least approximate normality when the population standard deviations of the two groups are unknown and the sample standard deviations have to be used as estimates.
The one sample z-test and one sample t-test are suitable for comparing the mean of a group with a constant, with the z-test being used under the assumption of at least approximate normality when the population standard deviation of the group is known, and the t-test being used under the assumption of at least approximate normality when the population standard deviation is unknown and the sample standard deviation has to be used as estimate.
The chi-square test can be used for determining whether there is a relationship between two categorical variables, each being present at two or more levels.
Thus, the correct answer is chi-square test.
b. Type I error:
In a hypothesis testing problem, a Type I error occurs when the test rejects the null hypothesis when it is true. In other words, if the null hypothesis is actually true, but the test finds evidence to support the alternative hypothesis, then there is said to be a Type I error.
Type II error:
In a hypothesis testing problem, a Type II error occurs when the test fails to reject the null hypothesis even though it is false. In other words, if the null hypothesis is not true and the alternative hypothesis is true, but the test fails to find evidence to support the alternative hypothesis, then there is said to be a Type II error.
In this case, a Type I error will occur, if there is really no relationship between handedness and penmanship, that is, if handedness and penmanship are independent, but the hypothesis test results in the conclusion that there is a significant relationship between handedness and penmanship.
Again, a Type II error will occur, if there really is a relationship between handedness and penmanship, but the hypothesis test fails to find the evidence of any relationship between handedness and penmanship, concluding that they are independent.

c. The chi-square test uses the observed frequencies of the data (like the frequencies given in the table in the question) and the expected frequencies (the frequencies expected under the assumption that the categorical variables are independent), to construct the test statistic. This test then aims to find out how close the observed frequencies are to the expected frequencies, and how likely is that closeness simply due to chance.NSIn other words, the aim of the chi-square test is to find out how well the observed frequencies fit the data, when it is expected that the categorical variables are independent.