I don't understand this at all..
Assume that the weights of Chinook Salmon in the Columbia River are normally distributed. You randomly catch and weigh 40 such salmon. The mean weight from your sample is 24.1 pounds with a standard deviation of 3.5 pounds. Test the claim that the mean weight of Columbia River salmon is greater than 23 pounds. Test this claim at the 0.10 significance level.
(a) What is the test statistic? Round your answer to 2 decimal places.
t x =
(b) What is the critical value of t? Use the answer found in the t-table or round to 3 decimal places.
tα =
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
To test the claim, we need to perform a one-sample t-test. Let's break down the process of finding the test statistic (t) and the critical value of t.
(a) Test Statistic:
The test statistic for a one-sample t-test is calculated using the formula:
t = (x̄ - μ) / (s / √n)
In this formula:
- x̄ represents the sample mean (24.1 pounds)
- μ represents the claimed population mean (23 pounds)
- s represents the sample standard deviation (3.5 pounds)
- n represents the sample size (40)
Plugging in the values, we get:
t = (24.1 - 23) / (3.5 / √40)
Calculating this expression, we find:
t = 1.1 / (3.5 / √40) ≈ 0.37 (rounded to 2 decimal places)
Therefore, the test statistic (t) is approximately 0.37.
(b) Critical Value of t:
The critical value of t can be found from the t-table or using statistical software. Since the significance level is given as 0.10, it implies that the α value (probability of Type I error) is 0.10.
With a one-tailed test (testing if the mean weight is greater than 23 pounds), we need to find the critical value of t for a 0.10 significance level and degrees of freedom (df) equal to (n - 1).
In this case, df = 40 - 1 = 39.
Using the t-table or statistical software, the critical value at α = 0.10 for df = 39 is approximately 1.314 (rounded to 3 decimal places).
Therefore, the critical value of t (tα) is approximately 1.314.
In summary:
(a) The test statistic (t) is approximately 0.37 (rounded to 2 decimal places).
(b) The critical value of t (tα) is approximately 1.314 (rounded to 3 decimal places).