Question

in a statistics activity, students are asked to spin a penny and a dime and determine the proportion of times that each lands with tails up. the students believe that since a dime is lighter, it will have a lower proportion of times landing tails up compared with the penny. the students are instructed to spin the penny and the dime 30 times and record the number of times each lands tails up. for one student, the penny lands tails side up 18 times, and the dime lands tails side up 20 times. let ( p_{d} ) = the true proportion of times a dime will land tails up and ( p_{p} ) = the true proportion of times a penny will land tails up. the ( p )-value for this significance test is 0.296. which of the following is the correct conclusion for this test of the hypotheses ( h_{0}: p_{d}-p_{p}=0 ) and ( h_{a}: p_{d}-p_{p}<0 ) at the ( alpha = 0.05 ) level?
the student should reject the null hypothesis since ( 0.296>0.05 ). there is sufficient evidence that the true proportion of times a dime will land tails up is significantly less than the penny.
the student should reject the null hypothesis since ( 0.296>0.05 ). there is insufficient evidence that the true proportion of times a dime will land tails up is significantly less than the penny.
the student should fail to reject the null hypothesis since ( 0.296>0.05 ). there is sufficient evidence that the true proportion of times a dime will land tails up is significantly less than the penny.
the student should fail to reject the null hypothesis since ( 0.296>0.05 ). there is insufficient evidence that the true proportion of times a dime will land tails up is significantly less than the penny.

Explanation:

In hypothesis testing, if the \(P -\)value is greater than the significance level \(\alpha\), we fail to reject the null hypothesis. Here, \(\alpha = 0.05\) and \(P - value=0.296\). Since \(0.296>0.05\), we fail to reject \(H_0:p_D - p_P=0\). Failing to reject the null hypothesis means there is insufficient evidence to support the alternative hypothesis \(H_a:p_D - p_P<0\) (which claims that the proportion of times a dime lands tails up is less than that of a penny).

Answer:

The student should fail to reject the null hypothesis since \(0.296 > 0.05\). There is insufficient evidence that the true proportion of times a dime will land tails up is significantly less than the penny.