Complete the hypothesis test: null hypothesis = 52, alternative hypothesis =<52, a=0.01 usin data set 45,47,46,58,59,49,46,54,53,52,47,41
t = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√(n-1)
Since only one SD can be calculated, you can use just that to determine SEdiff.
Consult your t-test table in the back of your stat book to determine the probability for df = 11.
To complete the hypothesis test, we can follow these steps:
Step 1: State the null and alternative hypotheses:
The null hypothesis (H0) is a statement of no effect or no difference. In this case, the null hypothesis is H0: μ = 52, where μ represents the population mean.
The alternative hypothesis (H1) is the statement we want to test, often seen as the opposite of the null hypothesis. In this case, the alternative hypothesis is H1: μ ≤ 52, indicating that the population mean is less than or equal to 52.
Step 2: Determine the level of significance (α):
The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is true. In this case, α = 0.01, which means we are willing to accept a 1% chance of making a Type I error (rejecting the null hypothesis when it is true).
Step 3: Collect and summarize the data:
Based on the provided dataset: 45, 47, 46, 58, 59, 49, 46, 54, 53, 52, 47, 41. We need to calculate the sample mean and sample standard deviation.
Sample mean (x̄) = (45 + 47 + 46 + 58 + 59 + 49 + 46 + 54 + 53 + 52 + 47 + 41) / 12 = 50.96
Sample standard deviation (s) = sqrt(((45-50.96)^2 + (47-50.96)^2 + (46-50.96)^2 + (58-50.96)^2 + (59-50.96)^2 + (49-50.96)^2 + (46-50.96)^2 +(54-50.96)^2 + (53-50.96)^2 + (52-50.96)^2 + (47-50.96)^2 + (41-50.96)^2) / 11) = 5.905
Step 4: Conduct the hypothesis test:
To conduct the hypothesis test, we can use the t-distribution since the population standard deviation is unknown.
Calculate the test statistic (t):
t = (x̄ - μ) / (s / √n)
t = (50.96 - 52) / (5.905 / √12)
t ≈ -0.419
Step 5: Determine the critical region or p-value:
Since the alternative hypothesis is one-tailed (μ ≤ 52), we will use the t-distribution's critical value or p-value corresponding to the desired level of significance (α = 0.01).
To find the critical value:
Using a t-table or a statistical software, find the critical t-value for 11 degrees of freedom (n-1) at α = 0.01. In this case, the critical t-value is approximately -2.718.
Alternatively, we can calculate the p-value using the t-distribution:
p-value = P(T ≤ t)
Using a t-table or a statistical software, find the probability of observing a t-value less than or equal to -0.419. The resulting p-value is approximately 0.343.
Step 6: Make a decision:
If our test statistic t is within the critical region (less than the critical t-value) or if our p-value is less than α, we can reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, -0.419 is not less than the critical t-value of -2.718, and the p-value of 0.343 is greater than α = 0.01.
Therefore, we fail to reject the null hypothesis.
Step 7: Draw a conclusion:
Based on the analysis, there is not enough evidence to support the claim that the population mean is less than or equal to 52. The dataset does not provide conclusive evidence to reject the null hypothesis.