A two-tailed hypothesis test at the 5 percent level of significance (£\ = .05) is exactly equivalent to asking whether the 95 percent confidence interval for the mean includes the hypothesized mean. If the confidence interval includes the hypothesized mean
H0: £g = 216, then we cannot reject the null hypothesis. Also, £m = .023
In this case the 95 percent confidence interval would be
¡x ¡Ó z£m¡Ôn = 216.007 ¡Ó 1.96 (.023)¡Ô50 = 216.001 < £g < 216.013
Since this confidence interval does not include 216, we reject the null hypothesis H0: £g = 216.
However, the decision is rather a close one as it was with the two-tailed hypothesis test, since the lower limit of the confidence interval almost includes 216.
Interpretation: Although the sample mean 216.007 might seem very close to 216, it is
more than two standard deviations from the desired mean. This example shows that even a small difference can be significant. It all depends on £m and n, that is, on the standard error of the mean in the denominator of the test statistic. In this case, there is a high degree of precision in the manufacturing process (£m = .023 is very small) so the standard error (and hence the allowable variation) is extremely small. Such a tiny difference in means would not be noticeable to consumers, but stringent quality control standards are applied to ensure that no shipment goes out with any noticeable non-conformance.
Please provide responses to the following:
(a) Give a brief example of a two-tailed hypothesis.
(b) Small differences can be statistical significant e.g. a high degree of precision in a manufacturing process. Give another example of where we can have such statistical significance with small differences.
(c) List the steps in the formal hypothesis testing process. Which step do you feel will be the most challenging for you? What does this challenging step involve?
For the School Subject, what on earth is Res? I thought it asked for a resumé and that's the only reason I opened it! If you correctly label the School Subject, you will get the correct teacher faster.
Sra
(a) A brief example of a two-tailed hypothesis can be testing whether there is a difference in the means of two groups. For example, let's say we want to investigate whether there is a difference in the average height between male and female students. The null hypothesis (H0) would be that there is no difference in the average height, while the alternative hypothesis (Ha) would be that there is a difference in the average height.
(b) Another example where we can have statistical significance with small differences is in medical studies. Let's say a new drug is being tested to see if it is effective in reducing blood pressure. Even a small decrease in blood pressure could be statistically significant if the sample size is large enough. In this case, the null hypothesis would be that the drug has no effect on blood pressure, while the alternative hypothesis would be that the drug does have an effect.
(c) The steps in the formal hypothesis testing process are as follows:
1. Identify the research question: Clearly define the question or problem being investigated.
2. Formulate the null and alternative hypotheses: State the null hypothesis (H0) and the alternative hypothesis (Ha) in terms of population parameters.
3. Choose the significance level: Determine the desired level of significance (£\) for the hypothesis test. This determines the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true.
4. Collect and analyze the data: Gather the necessary data and analyze it using appropriate statistical methods.
5. Calculate the test statistic: Calculate the test statistic based on the sample data and the chosen test.
6. Determine the critical region: Identify the critical region(s) based on the significance level and the test statistic. This is the region(s) of the test statistic values that would lead to rejecting the null hypothesis.
7. Make a decision: Compare the test statistic to the critical region(s) and make a decision about whether to reject or fail to reject the null hypothesis.
8. Interpret the results: Interpret the results in the context of the research question and make conclusions about the hypothesis being tested.
The step that might be challenging for me is step 5, which involves calculating the test statistic. This step requires a good understanding of the statistical method being used and the calculations involved. It can be complex depending on the specific hypothesis test being performed.