Answer: To determine what the researchers should do, we need to perform a hypothesis test to assess whether the sample mean of 58.5 lbs significantly differs from the population mean of 57 lbs.
The hypothesis test can be set up as follows:
Null Hypothesis (H0): The sample mean is equal to the population mean.
Alternate Hypothesis (H1): The sample mean is significantly different from the population mean.
Mathematically, this can be represented as:
H0: u = 57 (where μ is the population mean)
H1: u ≠ 57
Next, we need to calculate the test statistic and compare it to the critical value (from the t-distribution) for the chosen level of significance (commonly 0.05 or 0.01).
The test statistic is given by:
t = (x - u) / (s/√n)
where x is the sample mean, u is the population mean, s is the sample standard deviation, and n is the sample size.
Given values:
x = 58.5 lbs (sample mean)
u = 57 lbs (population mean)
s = 6.3 lbs (population standard deviation)
n = 45 (sample size)
Now, let's calculate the test statistic:
t = (58.5 - 57) / (6.3/√45)
t = 1.5 / (6.3/√45)
t ≈ 1.5 / 0.9404
t ≈ 1.5951
With the test statistic calculated, the researchers should consult a t-distribution table or use statistical software to find the critical value corresponding to their chosen level of significance (e.g., α = 0.05).
If the absolute value of the test statistic is greater than the critical value, then the researchers can reject the null hypothesis, indicating that the sample mean is significantly different from the population mean. If the absolute value of the test statistic is less than the critical value, then the researchers cannot reject the null hypothesis, and there is not enough evidence to conclude that the sample mean is significantly different from the population mean.
Therefore, the researchers should compare the calculated test statistic (1.5951) to the critical value and make a decision accordingly.