Question

According to a research study, people are able to better manage their time when they wear watches than when they use mobile phones or other gadgets to keep track of time. To test this, 300 college students were randomized into a treatment group and a control group. The treatment group was asked to wear a watch every day, and the control group was asked to avoid wearing a watch for 50 days. The number of occasions when students from both groups arrived later than their scheduled time was noted. The results of the experiment showed that the mean score of the treatment group was exactly 15 points less than the mean score of the control group. To test whether the results could be explained by random chance, the researchers created the following table, which summarizes the results of 1,000 randomizations of the data with differences of means rounded to the nearest 5 points. Consider the significance level to be set at 5%, so results lower than 5% can be considered statistically significant.
Question: Determine the probability of the treatment group’s mean being lower than the control group’s mean by 15 points or more.

Answer 1

Answer:

$P(X \leq -15) = 7.8\%$

There is a 7.8% probability that the treatment group’s mean is lower than the control group’s mean by 15 points or more.

Since the given significance level is 5% and the results lower than 5% are considered statistically significant, therefore, the obtained result of 7.8% is not statistically significant because it is not less than 5%.

Step-by-step explanation:

According to a research study, people are able to better manage their time when they wear watches than when they use mobile phones or other gadgets to keep track of time.

To test this, 300 college students were randomized into a treatment group and a control group.

The treatment group was asked to wear a watch every day, and the control group was asked to avoid wearing a watch for 50 days.

The number of occasions when students from both groups arrived later than their scheduled time was noted.

The results of the experiment showed that the mean score of the treatment group was exactly 15 points less than the mean score of the control group.

To test whether the results could be explained by random chance, the researchers created the following table, which summarizes the results of 1,000 randomizations of the data with differences of means rounded to the nearest 5 points.

Consider the significance level to be set at 5%, so results lower than 5% can be considered statistically significant.

We are asked to find the probability of the treatment group’s mean being lower than the control group’s mean by 15 points or more.

$P(X \leq -15) = ?$

From the given table,

Treatment Group Mean - Control Group Mean       |      frequency

-25                                                                              |          10

-20                                                                               |          18

-15                                                                                |          50  

Frequency of less than or equal to -15 points = 10 + 18 + 50 = 78

Total frequency = 1000

So the probability is given by

$P(X \leq -15) = \frac{78}{1000} \\\\P(X \leq -15) = 0.078 \\\\P(X \leq -15) = 7.8\%$

Therefore, there is a 7.8% probability that the treatment group’s mean is lower than the control group’s mean by 15 points or more.

Since the given significance level is 5% and the results lower than 5% are considered statistically significant, therefore, the obtained result of 7.8% is not statistically significant because it is not less than 5%.