a restaurant that bills its house account monthly is concerned that the average monthly bill exceeds $200 per account. A random sample of twelve account is selected, resulting in a sample mean of $220 and a standard deviation of $12. Use a= 0.05. what is the critical value for this test? show the rejection and non rejection area. calculate the test statistic?
To solve this problem, we need to perform a hypothesis test using the given data. Here are the steps:
1. State the null and alternative hypotheses:
- Null hypothesis (H₀): The average monthly bill does not exceed $200 per account.
- Alternative hypothesis (H₁): The average monthly bill exceeds $200 per account.
2. Determine the significance level: The given significance level is α = 0.05.
3. Find the critical value: The critical value is determined by the chosen significance level and the type of test being performed. Since it is a one-tailed test (the alternative hypothesis specifies "exceeds"), we need to find the critical value corresponding to the rejection area in the upper tail of the distribution.
The critical value can be obtained using a statistical table for the t-distribution with n-1 degrees of freedom. Since the sample size is 12 (n = 12), the degrees of freedom will be 12 - 1 = 11.
For α = 0.05 and 11 degrees of freedom, the critical value is approximately tₐ = 1.796. (You can find this by looking up the value in a t-distribution table or using a statistical calculator.)
Rejection area: The rejection area is in the upper tail of the distribution, corresponding to t > tₐ (t-value greater than the critical value). In this case, it means if the test statistic (t-value) exceeds 1.796, we will reject the null hypothesis.
Non-rejection area: The non-rejection area is in the lower tail and the middle region of the distribution, corresponding to t ≤ tₐ (t-value less than or equal to the critical value). In this case, it means if the test statistic (t-value) is less than or equal to 1.796, we will not reject the null hypothesis.
4. Calculate the test statistic (t-value):
The formula to calculate the t-value is: t = (sample mean - hypothesized mean) / (sample standard deviation / √n)
Given data:
Sample mean (x̄) = $220
Sample standard deviation (s) = $12
Sample size (n) = 12
Hypothesized mean (μ) = $200
Plugging these values into the formula:
t = (220 - 200) / (12 / √12) = 20 / (12 / √12) = 20 / (12 / 3.4641) = 20 / 3.4641 ≈ 5.7735
Therefore, the calculated test statistic (t-value) is approximately 5.7735.
To summarize:
- Critical value: tₐ = 1.796
- Rejection area: t > 1.796
- Non-rejection area: t ≤ 1.796
- Test statistic: t ≈ 5.7735