Question

A student wants to assess whether her dog Groot tends to chase her blue ball more often than she chases her red ball. The student rolls both a blue ball and a red ball at the same time and observes which ball Groot chooses to chase. Repeating this process a total of 96 times, the student finds that Groot chased the blue ball 52 times and the red ball 44 times.

1. What are the observational units in this study?

2. What is the variable in this study (there is only one)?

3. State the appropriate null and alternative hypotheses, in both words and in symbols.

4. Calculate the test statistic and determine the p-value.

5. Decide if you reject the null hypothesis at the 0.10 significance level. Explain your decision.

6. Write a one sentence conclusion, summarizing what the data reveal about whether Groot tends to chase the blue ball more often than the red ball.

7. Would the p-value have been larger, smaller, or the same if a larger sample size had been used and all else turned out the same? (Do not bother to explain.)

8. Would the p-value have been larger, smaller, or the same if the level of significance used had been smaller and all else turned out the same? (Do not bother to explain.)

Answer 1

1. The student wants to assess whether her dog Groot tends to chase her blue ball more often than she chases her red ball. Hence, the observational unit is the color of the ball chosen, by Groot, to chase.

2. The variable in the study is:

X = 1, Color of the ball chosen to chase is Blue

= 0, otherwise (when red)

3. We find that the variable has two possible outcomes; with the outcome of interest - Chasing a Blue ball as the event of success. Also, the same (independent) trial is repeated n = 96 times.

We may say that

where, No. of trials = n = 96 and

Probability of success = p = 52 / 96 = 0.54

We may hypothesize the that the proportion of blue ball and red ball chosen are the same: Equal proportion 0.50 (p0 say)

Vs   

H0: There is no difference in the proportion of Blue and Red balls chased. Vs Ha: Proportion of Blue balls chased is greater than that of Red balls.

4. The appropriate statistical test to test the above hypothesis would be a One sample test for Proportion.But before running the test we must first ensure whether all the assumptions of the test are satisfied. The assumptions of One sample test for Proportion are as follows:

· The data is randomly collected from the population

· The underlying distribution of the population is binomial distribution.

· There are only two possible outcomes in each trial

· When np ≥ 5 and np(1 – p) ≥ 5, the binomial distribution can be approximated by the normal distribution.

Assumping that all the above are satisfied, we may go for computing the test statistic, given by the formula

Substituting the given values,

= 0.79

P-value can be determined as:

From standard normal table,

= 1 - 0.78524

= 0.2148

5. Since, the p-value of the test 0.2148 > 0.10 is not significant, we fail to reject the null at 10% level. This is because, the p-value of the test, i.e the probability of obtaining a result as extreme as the one obtained, when the null hypothesis is true is quite high (> 0.10 fixed sig. level). Hence, there is greater chance that the null might actually be true, which leads to the decision of failing to reject the same.

6. We may conclude that the data does not provide sufficient evidence to support the claim that the student's dog Groot tends to chase her blue ball more often than she chases her red ball.

7. If all the observations remained same except for the sample size, we may look at the test statistic Z - As n increases, denominator decreases and in turn, Z increases. Larger Z would produce more extreme results, thus, reducing the p-value.

P- value decreases as sample size increases

8. If the sig. level is fixed at a lower level, the p-value would still remain the same.

Eg. If alpha = 5%, the p-value would still be 0.2148 and the decision, would also be the same : P value = 0.2148 > 0.05.