Please complete the following 5-step hypothesis testing procedure in your Learning Teams:
A simple random sample of forty 20-oz bottles of Coke® from a normal distribution is obtained and each bottle is measured for number of ounces of Coke® in the bottle. The sample mean is 19.4 oz with standard deviation of 1.8 oz. Test the claim at an alpha significance level of 5% that on average there is exactly 19.8 oz of Coke® in each bottle.
To complete the 5-step hypothesis testing procedure for this problem, follow these steps:
Step 1: State the hypotheses.
The null hypothesis (H0) is that on average there is exactly 19.8 oz of Coke in each bottle. The alternative hypothesis (Ha) is that the average amount of Coke in each bottle is not equal to 19.8 oz.
H0: μ = 19.8
Ha: μ ≠ 19.8
Step 2: Formulate an analysis plan.
In this step, we specify the alpha level (significance level) and the criteria for rejecting the null hypothesis. In this case, the alpha level is given as 5%, which means we will reject the null hypothesis if the probability of observing the sample mean is less than 5%.
Step 3: Analyze the sample data.
Calculate the test statistic and the corresponding p-value.
The test statistic (t-score) can be calculated using the formula:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
Using the given information:
sample mean (x̄) = 19.4 oz
hypothesized mean (μ) = 19.8 oz
sample standard deviation (s) = 1.8 oz
sample size (n) = 40
Substituting these values into the formula, we get:
t = (19.4 - 19.8) / (1.8 / sqrt(40))
Step 4: Interpret the results.
Compare the calculated test statistic to the critical value(s) from the t-distribution table or use statistical software to determine the p-value. If the p-value is less than the significance level (alpha), we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
Step 5: State the conclusion.
Based on the results obtained in Step 4, state the conclusion in terms of the original claim. If the null hypothesis is rejected, it means there is strong evidence to support the alternative hypothesis. If the null hypothesis is not rejected, it means there is not enough evidence to support the alternative hypothesis.
Note: In this case, since the alternative hypothesis is two-tailed (μ ≠ 19.8), we will need to compare the p-value to the significance level (α) divided by 2 (2-tailed test).
Using the calculated test statistic, degrees of freedom, and the significance level, you can find the p-value from the t-distribution table or by using statistical software. If the p-value is less than α/2, we reject the null hypothesis.
By following these steps, you can complete the 5-step hypothesis testing procedure for this problem and determine whether to reject or fail to reject the null hypothesis.
Step 1: State the null and alternative hypotheses
The first step in hypothesis testing is to state the null and alternative hypotheses.
- Null hypothesis (H0): The average number of ounces of Coke in each bottle is 19.8.
- Alternative hypothesis (Ha): The average number of ounces of Coke in each bottle is not 19.8.
Step 2: Determine the test statistic
For this hypothesis test, we will use the t-test because the population standard deviation is unknown, and the sample size is relatively small (n < 30). The formula for the t-test statistic is:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
In this case:
- Sample mean (x̄) = 19.4 oz
- Hypothesized mean (μ) = 19.8 oz
- Sample standard deviation (s) = 1.8 oz
- Sample size (n) = 40
t = (19.4 - 19.8) / (1.8 / sqrt(40))
Step 3: Determine the critical value(s)
Since our significance level is 5% (alpha = 0.05), we need to determine the critical value for a two-tailed test. For a two-tailed test at alpha = 0.05 and degrees of freedom (df) = n - 1, the critical t-values can be obtained from a t-table or a t-distribution calculator. In this case, with df = 39, the critical t-values are approximately ±2.024.
Step 4: Calculate the test statistic value
Calculate the test statistic value using the formula from step 2.
t = (19.4 - 19.8) / (1.8 / sqrt(40))
Step 5: Make a decision
Compare the calculated test statistic value with the critical value(s) obtained in step 3. If the calculated test statistic value falls outside the critical region(s), we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
In this case, if the calculated t-value falls outside the range of -2.024 to 2.024, we reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis. Otherwise, if the calculated t-value falls within the range of -2.024 to 2.024, we fail to reject the null hypothesis and conclude that there is not enough evidence to support the alternative hypothesis.