Consider the following Hypotheses test:
Ho:m>=80
Ha:m<80
A sample of 121 provided a sample mean of 77.3. The population Standard Deviation is known to be 16.5.
A. compute the value of test statistic
B. Determin the p value: and at 93.7% confidence, Test the above hypothesis
C. using the critcal value approach at 93.7 % confidence, test the hypotisis
To solve this hypothesis test, we can follow the steps for each question:
A. Compute the value of the test statistic:
The test statistic we need to calculate is the Z-score, which measures how many standard deviations an observed sample mean is away from the hypothesized population mean.
The formula to calculate the Z-score is:
Z = (X - μ) / (σ / √n)
Given information:
Sample mean (X) = 77.3
Population mean (μ) = 80 (from Ho)
Population standard deviation (σ) = 16.5
Sample size (n) = 121
Plugging the values into the formula, we get:
Z = (77.3 - 80) / (16.5 / √121)
Z = -2.7 / (16.5 / 11)
Z = -2.7 / 1.5
Z ≈ -1.8
So, the value of the test statistic is approximately -1.8.
B. Determine the p-value and test the above hypothesis at 93.7% confidence:
To determine the p-value, we need to find the probability of observing a test statistic as extreme as the one calculated or more extreme, assuming the null hypothesis is true.
Since the alternative hypothesis is Ha: m < 80, we need to find the probability of obtaining a Z-score less than or equal to -1.8.
Using a Z-table or calculator, we find the area under the standard normal distribution curve to the left of Z-score -1.8 is approximately 0.0359.
This probability is the p-value.
To test the hypothesis at 93.7% confidence level, we compare the p-value with the significance level (1 - confidence level).
1 - 0.937 = 0.063
Since the p-value (0.0359) is less than the significance level (0.063), we reject the null hypothesis Ho. There is enough evidence to support the alternative hypothesis Ha: m < 80 at the 93.7% confidence level.
C. Using the critical value approach at 93.7 % confidence, test the hypothesis:
To use the critical value approach, we need to compare the calculated test statistic with the critical value from the standard normal distribution, considering the confidence level.
The critical value corresponds to the value of Z that leaves a specified area to the left of it, equal to the chosen confidence level.
At 93.7% confidence level, we find the critical value by finding the Z-score that leaves 93.7% to the left under the standard normal distribution curve. This corresponds to an area of 1 - 0.937 = 0.0637.
Using a Z-table or calculator, we find that the critical value is approximately -1.57.
Comparing the calculated test statistic (-1.8) with the critical value (-1.57):
Since the test statistic is less extreme (more negative) than the critical value, we again reject the null hypothesis Ho. There is enough evidence to support the alternative hypothesis Ha: m < 80 at the 93.7% confidence level.