assignment 1

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Each player is dealt a hand of five cards from a standard deck of 52 cards. One player records the number of Hearts in each of the 170 hands he plays one night. The data are shown in the table below:

# of Hearts	0	1	2	3	4	5
# of Hands	33	72	49	13	3	0
Expected

We would like to conduct a chi-square goodness of fit test at the 10% level of significance to determine whether the number of Hearts per hand follows a binomial distribution with parameter p = 0.25. (a) Calculate all expected cell counts and enter them in the table above. (Round all values to two decimal places.) (b) The expected cell counts for 4 and 5 Hearts are both less than 5, so we must merge them with the cell for 3 Hearts. Enter all appropriate expected counts in the table below. Then calculate the cell chi-square values and enter them in the table. (Round all values to two decimal places.)

# of Hearts	0	1	2
# of Hands	33	72	49	16
Expected
Cell Chi-Square

(c) The value of the test statistic for the appropriate test of significance (rounded to two decimal places) is (d) The P-value for the appropriate test of significance is:

between 0.01 and 0.02.

between 0.02 and 0.025.

between 0.025 and 0.05.

between 0.05 and 0.10.

between 0.10 and 0.15.

between 0.15 and 0.20.

between 0.20 and 0.25.

greater than 0.25.

(e) Suppose we had instead used the critical value method to conduct the test. The decision rule would be to reject H0 if . (f) What is the correct conclusion for this test?

Fail to reject H0. There is insufficient evidence that the number of Hearts per hand does not follow a binomial distribution with parameter p = 0.25.

Fail to reject H0. There is sufficient evidence that the number of Hearts per hand follows a binomial distribution with parameter p = 0.25.

Reject H0. There is insufficient evidence that the number of Hearts per hand does not follow a binomial distribution with parameter p = 0.25.

Fail to reject H0. There is sufficient evidence that the number of Hearts per hand does not follow a binomial distribution with parameter p = 0.25.

Fail to reject H0. There is insufficient evidence that the number of Hearts per hand follows a binomial distribution with parameter p = 0.25.

Reject H0. There is sufficient evidence that the number of Hearts per hand does not follow a binomial distribution with parameter p = 0.25.

Reject H0. There is sufficient evidence that the number of Hearts per hand follows a binomial distribution with parameter p = 0.25.

Reject H0. There is insufficient evidence that the number of Hearts per hand follows a binomial distribution with parameter p = 0.25.

(g) Which of the following statements must be true?

We have made a Type I Error in our conclusion.

We have made a Type II Error in our conclusion.

We have not made a Type I or a Type II Error. Our conclusion is correct.

It is impossible to know whether we have made an error in our conclusion.

(h) Suppose we only wanted to test whether the number of Hearts per hand follows a binomial distribution (without specifying a particular value of the parameter p). The estimated value of the parameter p (rounded to four decimal places) would be .

(i) If the number of Hearts had to be estimated, then, using the critical value method, the decision rule would be to reject H0 if