Hypothesis testing on a paired sample, using the sign test.
An educational psychologist suspects that having cats improves performance on statistics finals. With funding from the Socks foundation, she did a pair study, with carefully chosen pairs of students, one having a cat, and the other none. She obtained thae following faired data values, consiting of: test score of student with cat/test score of student without cat
58/56, 51/51, 66/65, 56/55, 67/66, 77/45, 88/60, 61/60, 91/92, 92/88, 100/54, 100/100, 88/76, 99/95, 67/57, 57/86, 94/88, 89/88
Explain a way of dealing with ties and find the p-value.
Here are a few hints.
Step 1: Look at the values of each set of test scores (test score of student with cat versus test score of student without cat). Assign a minus if the scores decreased; assign a plus if the scores increased. Assign a 0 for those who remained the same.
Step 2: Use the appropriate table for this type of test. Determine the total number of the least frequent sign. Remember that this is a one-tailed test. Determine the p-value from the table. (Hint: n = 16; the 0's are not counted.)
I hope this will help.
To conduct hypothesis testing using the sign test with paired samples, you need to compare the differences between the paired observations. In this case, the differences are the test scores of students with cats minus the test scores of students without cats.
To deal with the ties in the data (i.e., cases where the test scores are the same for both students in a pair), the sign test uses a specific approach. You will ignore those tied pairs and focus only on the pairs with distinct differences.
Let's calculate the differences based on the given data:
58 - 56 = 2
51 - 51 = 0 (tie, ignore)
66 - 65 = 1
56 - 55 = 1
67 - 66 = 1
77 - 45 = 32
88 - 60 = 28
61 - 60 = 1
91 - 92 = -1
92 - 88 = 4
100 - 54 = 46
100 - 100 = 0 (tie, ignore)
88 - 76 = 12
99 - 95 = 4
67 - 57 = 10
57 - 86 = -29
94 - 88 = 6
89 - 88 = 1
Now we have a smaller set of differences that we will use for the sign test. Next, we assign signs (+ or -) to each difference. If the difference is positive, assign a plus (+) sign; if it's negative, assign a minus (-) sign.
+
0 (tie, ignore)
+
+
+
+
+
+
+
-
+
-
0 (tie, ignore)
+
+
+
-
+
+
Now, we can calculate the p-value using the sign test. The p-value represents the probability of observing a result as extreme as or more extreme than the one obtained, assuming the null hypothesis is true.
To find the p-value, we count the number of positive signs (n+) and the total number of pairs excluding ties (N). In this case, n+ is 12 and N is 15.
We need to calculate the p-value based on the binomial distribution. If the null hypothesis is true (having a cat does not improve test scores), the probability of observing a positive difference is 0.5.
Using the binomial distribution, we can calculate the probability of observing 12 or more positive differences out of 15 pairs with a probability of success of 0.5. This gives us the p-value.
There are multiple ways to calculate the p-value based on the binomial distribution. One way is to use a statistical software package or an online calculator specifically designed for hypothesis testing.
Alternatively, you can use the cumulative binomial probability formula and calculate the p-value manually by summing up the probabilities for 12, 13, 14, and 15 positive differences out of 15 pairs. However, this approach requires knowledge of binomial probabilities and extensive calculations.
To get the exact p-value for this specific dataset and hypothesis, it is recommended to use statistical software or an online calculator that can provide accurate results efficiently.