Question

Using Probability to Form Conclusions. In Exercises 37–40, use the given probability value to determine whether the sample results could easily occur by chance, then form a conclusion.

Coffee Talk A study on the enhancing effect of coffee on long-term memory found that 35 participants given 200 mg of caffeine performed better on a memory test 24 hours later compared to the placebo group that received no caffeine.

a. There was a probability of 0.049 that the difference between the coffee group and the placebo group was due to chance. What do you conclude?

b. A group given a higher dose of 300 mg performed better than the 200 mg group, with a probability of 0.75 that this difference is due to chance. What do you conclude?

Short answer

(a)

The difference between the 200 mg coffee group and the placebo does not occur by chance. It can be concluded that a dose of 200 mg has a significant effect on memory.

(b)

The 300 mg coffee group performed better than the 200 mg coffee group by chance.

Thus, the dose of 300 mg coffee does not have a significantly better effect on memory than a dose of 200 mg coffee.

Step-by-step solution

Given information

The group with no caffeine did not perform better on a memory test than the group of 35 participants that took 200 mg of caffeine.

The group given a dose of 300 mg performed better than a group given 200 mg, displaying a probability of 0.75 by chance.

Probability analysis

Theprobability value is the value between 0 and 1 that gives an idea of the degree of possibility of occurrence of an event. A value lower than 0.05 would imply that the event is unusual or unlikely to occur.

Interpret the probability value to establish the difference between two groups

(a)

It is known that the probability of getting the difference between the memory test for two groups by chance is 0.049.

The probability that the difference between memory test results for the 200 mg coffee group and the placebo group has occurred due to chance is 0.049.

This is lower than 0.05. Thus, it can be said that the difference has not occurred by chance.

Therefore, it can be concluded that a dose of 200 mg coffee has a significant effect on memory.

Interpret the probability for the difference between two groups 

(b)

It is required to evaluate the difference in memory test results for two groups:

• 300 mg dose of caffeine
• 200 mg dose of caffeine

The probability that the 300 mg dose performed better by chance is 0.75.

The probability that the difference between the 300 mg coffee group and the 200mg group on memory has occurred due to chance is equal to 0.75.

The value is greater than 0.05.

Thus, it can be said that the 300 mg dose group performed better than the 200 mg dose group by chance.

Therefore, it can be concluded that a dose of 300 mg coffee does not have a significantly better effect on memory than a dose of 200mg coffee.