A researcher is testing the hypothesis that consuming a sports drink during exercise improves endurance. A sample of n=50 male college students is obtained and each student is given a series of three endurance tasks and asked to consume 4 ounces of the drink during each break between tasks. The overall endurance score for this sample is M=53. For the general population, without any sports drink, the scores for this task average m=50 with a standard deviation of σ=12.
a. Can the researcher conclude that endurance scores with the sports drink are significantly higher than scores without the drink? Use a one- tailed test with α=.05.
b. Can the researcher conclude that endurance scores with the sports drink are significantly different than scores without the drink? Use a two- tailed test with α=.05.
c. You should find that the two tests lead to different conclusions. Explain why.
Briefly explain how increasing sample size influences each of the following. Assume that all other factors are held constant.
a. The size of the z- score in a hypothesis test.
b. The size of Cohen’s d.
c. The power of a hypothesis test.
To determine if the researcher can conclude that endurance scores with the sports drink are significantly higher than scores without the drink, we need to conduct a hypothesis test.
a. One-tailed Test:
Null hypothesis (H0): The mean endurance scores with the sports drink are not significantly different from the mean scores without the drink. μ = 50
Alternative hypothesis (H1): The mean endurance scores with the sports drink are significantly higher than the mean scores without the drink. μ > 50 (one-tailed)
Since we are given that the sample mean is 53, we can use the formula for a one-sample t-test:
t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))
t = (53 - 50) / (12 / sqrt(50))
t = 3 / (12 / 7.071)
t = 3 / 1.699
t = 1.764 (calculated t-value)
Next, we need to compare the calculated t-value to the critical t-value at α=0.05 (one-tailed) using a t-table or statistical software. For n=50, the degrees of freedom (df) is 49.
The critical t-value at α=0.05 (one-tailed) and df=49 is approximately 1.677.
Since the calculated t-value (1.764) is greater than the critical t-value (1.677), we can reject the null hypothesis. Therefore, the researcher can conclude that endurance scores with the sports drink are significantly higher than scores without the drink.
b. Two-tailed Test:
Null hypothesis (H0): The mean endurance scores with the sports drink are not significantly different from the mean scores without the drink. μ = 50
Alternative hypothesis (H1): The mean endurance scores with the sports drink are significantly different from the mean scores without the drink. μ ≠ 50 (two-tailed)
For a two-tailed test, we need to calculate the probability of observing a sample mean as extreme as 53 (or more extreme), given that the null hypothesis is true.
We can use the same formula as in the one-tailed test to calculate the t-value:
t = (53 - 50) / (12 / sqrt(50))
t = 3 / (12 / 7.071)
t = 3 / 1.699
t = 1.764 (calculated t-value)
Next, we need to compare the calculated t-value to the critical t-values at α=0.05/2 (two-tailed) using a t-table or statistical software. For n=50, the degrees of freedom (df) is 49.
The critical t-values at α=0.05/2 (two-tailed) and df=49 are approximately -2.010 and 2.010.
Since the calculated t-value (1.764) falls between the critical t-values (-2.010 and 2.010), we fail to reject the null hypothesis. Therefore, the researcher cannot conclude that endurance scores with the sports drink are significantly different from scores without the drink.
c. The two tests lead to different conclusions because the one-tailed test only checks for a significant increase in endurance scores with the sports drink, whereas the two-tailed test allows for the possibility of a significant decrease as well. In this particular scenario, the one-tailed test provides evidence to support the alternative hypothesis, but the two-tailed test does not provide strong enough evidence to reject the null hypothesis. Therefore, the conclusions differ depending on whether the analysis focuses on only an increase or considers any significant difference in both directions.
To determine whether the researcher can conclude that endurance scores with the sports drink are significantly higher than scores without the drink, we need to conduct hypothesis testing and perform statistical analysis. Here's how we can approach each part of the question:
a. One-Tailed Test:
To conduct a one-tailed test, we need to set up the null and alternative hypotheses. In this case, the null hypothesis (H0) would state that there's no significant difference between endurance scores with and without the sports drink. The alternative hypothesis (Ha) would state that endurance scores with the sports drink are significantly higher than scores without the drink.
H0: μ ≤ 50 (There is no significant difference)
Ha: μ > 50 (There is a significant difference)
We can use a one-tailed t-test to compare the sample mean (M=53) to the population mean (m=50) with a one-tailed significance level α = 0.05. By comparing the calculated t-value to the critical value from the t-distribution table, we can determine whether we reject or fail to reject the null hypothesis.
b. Two-Tailed Test:
To conduct a two-tailed test, we need to set up the null and alternative hypotheses. In this case, the null hypothesis (H0) would state that there's no significant difference between endurance scores with and without the sports drink. The alternative hypothesis (Ha) would state that endurance scores with the sports drink are significantly different than scores without the drink.
H0: μ = 50 (There is no significant difference)
Ha: μ ≠ 50 (There is a significant difference)
We can use a two-tailed t-test to compare the sample mean (M=53) to the population mean (m=50) with a two-tailed significance level α = 0.05. By comparing the calculated t-value to the critical values from the t-distribution table, we can determine whether we reject or fail to reject the null hypothesis.
c. Different Conclusions:
The two tests can lead to different conclusions because they have different alternative hypotheses.
In the one-tailed test, we are specifically testing whether endurance scores with the sports drink are significantly higher than scores without the drink. Therefore, we only consider the possibility of a significant difference in one direction (higher scores).
In the two-tailed test, we are testing whether endurance scores with the sports drink are significantly different than scores without the drink. This means we are considering the possibility of a significant difference in both directions (higher or lower scores).
The two tests have different levels of statistical power and different criteria for rejection of the null hypothesis. As a result, the conclusions may vary depending on the specific alternative hypothesis and the direction of the effect being investigated.