Question

There is a multiple choice question with 4 choices. Many students got the problem wrong. Out of a class of n=60 only 20 got the problem right (1/3 or p=.3333). Prof. K wants to determine if the
students answered the question correctly more frequently than chance, i.e., the null hypothesis is 1/4 or p=.25).

Does the success/failure criterion hold?

What conditions might invalidate the independence requirement?

i) students copying each other's answers

ii) different sections taking the test on different days

iii) students from different majors taking the test

iv) allowing a cheat sheet

What is the standard error for our sample estimate?

What is the lower bound of the confidence interval for the underlying population parameter at the 95% confidence level using a normal/Z distribution?

What is the upper bound of the confidence interval for the underlying population parameter, at the 95% confidence level using a normal/Z distribution?

Even though the null-hypothesis value (.25) is included in the confidence interval, we can be more precise by doing a hypothesis test. What is the standard error of the null hypothesis value?

What is the p-value of of the null hypothesis? Remember to double the p-value you get out of p-norm because this is a two-tailed test.

Based on this hypothesis test, we should...

i) fail to reject the alternative hypothesis

ii) accept the alternative hypothesis

iii) reject the null hypothesis

iv) accept the null hypothesis

v) fail to reject the null hypothesis

vi) reject the alternative hypothesis

We can think about this in yet another way: the chi-squared test. Let's say that the correct answer was 'c' and as mentioned earlier, the sample proportion is p=.3333. The sample proportion for 'a' is .0666667, for 'b' it is .3 and for 'd' it is .3.

What are the expected counts for each multiple-choice option under the null hypothesis?

What is the observed count of the 'a' option?

What is the observed count of the 'b' option?

What is the observed count of the 'c' option?

What is the observed count of the 'd' option?

What is the chi-squared statistic?

What is the p-value for the X^2 test?

Given these analyses, what is the best explanation we can draw, choosing from among the options below?

i) students guessed the answer at random

ii) 15 students knew the right answer and did not guess at random

iii) the guesses were not at random but there was confusion among some options

Answer 1

Note : Allowed to answer only 4 sub questions in one post.

Does the success/failure criterion hold?
Yes they hold because

What conditions might invalidate the independence requirement?

i) students copying each other's answers

Explanation : Independence criterion would require for the student to answer the test without any body's help. Hence only copying or cheat sheet would undermine it.

What is the standard error for our sample estimate?
0.0609

What is the lower bound of the confidence interval for the underlying population parameter at the 95% confidence level using a normal/Z distribution?
0.2139

What is the upper bound of the confidence interval for the underlying population parameter, at the 95% confidence level using a normal/Z distribution?
0.4527