2. A researcher hypothesized that the pulse rates of long-distance athletes differ from those of other athletes. He believed that the runners’ pulses would be slower. He obtained a random sample of 10 long-distance runners. He measured their resting pulses. Their pulses were 45, 45, 64, 50, 58, 49, 47, 55, 50, 52 beats per minute. The average resting pulse of athletes in the general population is normally distributed with a pulse rate of 60 beats per minute. What statistical test should be used to analyze the data
How about a one-sample t-test?
Usually you use z-tests when sample sizes are large (n is greater than
or equal to 30) whether or not you know the population standard deviation.
If you do not know the population standard deviation and have a small
sample (n < 30), then you can use t-tests.
To analyze the data and determine whether the pulse rates of long-distance athletes differ from those of other athletes, a statistical test called a t-test should be used. Specifically, a one-sample t-test is appropriate in this scenario because we have a sample mean (average resting pulse rates of long-distance runners) and want to compare it to a known population mean (average resting pulse rates of athletes in the general population).
To analyze the data and test whether the pulse rates of long-distance athletes differ from those of other athletes, a statistical test called the t-test should be used. Specifically, a one-sample t-test would be appropriate in this scenario.
A one-sample t-test compares the average pulse rates of the long-distance runners (the sample) to a known or hypothesized population mean (the average resting pulse of athletes in the general population, which is stated as 60 beats per minute).
The steps to conduct a one-sample t-test are as follows:
1. Define the null hypothesis (H0) and alternative hypothesis (Ha):
- H0: The average pulse rate of long-distance athletes is equal to 60 beats per minute.
- Ha: The average pulse rate of long-distance athletes is different from 60 beats per minute.
2. Calculate the sample mean (x̄) and sample standard deviation (s) from the given data:
- Sum of pulses = 45 + 45 + 64 + 50 + 58 + 49 + 47 + 55 + 50 + 52 = 515
- Sample mean (x̄) = Sum of pulses / Sample size = 515 / 10 = 51.5
- Calculate the sample standard deviation (s) using the formula for the sample standard deviation.
3. Determine the critical value(s) or find the p-value using the t-distribution:
- The critical value(s) depend on the desired significance level (e.g., α = 0.05) and the degrees of freedom (df = n - 1) in this case.
4. Calculate the t-statistic using the formula:
- t = (x̄ - μ) / (s / √n)
- x̄ is the sample mean
- μ is the hypothesized population mean
- s is the sample standard deviation
- n is the sample size
5. Compare the calculated t-statistic with the critical value(s) or p-value:
- If the calculated t-statistic falls into the critical region or the p-value is less than the chosen significance level, you reject the null hypothesis (H0).
- If the calculated t-statistic falls into the non-critical region or the p-value is greater than the chosen significance level, you fail to reject the null hypothesis (H0).
In summary, by conducting a one-sample t-test, you can determine if there is evidence to support the researcher's hypothesis that the pulse rates of long-distance athletes differ from those of other athletes.