(See attached file for full problem description)
1. What are the steps in formal hypothesis testing? Make at least two explanatory comments about each step.
Step 1: Formulate the null and the alternative hypotheses. The null hypothesis is designated H0 while the alternative hypothesis is identified as H1. The null should be tested in terms of some parameter value of the population. If strong evidence exists that the null hypothesis is incorrect, the alternative must be accepted. In a sense, you are actually testing the alternative hypothesis.
Step 2: Specify the level of significance. This information will assist in determining different hypotheses about the population parameter of interest. The z and t tests are frequently used as statistical tests, but each can test different data. For example, the z test is typically used when the n is greater than 30 while the t test is used where n < 30. In both tests, the data should be independent from each other, but the t test can be used in a paired-sample t test.
Step 3: Calculate the test statistic. Specify the sampling distribution, including all possible values of the test statistic obtained under the assumption that the null hypothesis is true. To test the hypothesis of µ when the standard deviation is known, the standard normal distribution is the desired sampling distribution.
Step 4: Define the region of rejection. Specify the alpha level as the probability of making a type I error, which represents the level of risk one is willing to take in rejecting Ho. Next, establish the critical rejection region on the sampling distribution that would define the type of sample data it would take to reject H0. This critical rejection region can be determined by establishing whether H1 is non-directional or directional.
Step 5: Select the appropriate hypothesis. The time has arrived to conduct the experiment by collecting data. A random sampling is taken and the test statistic obtained or the researcher can obtain a p-value. Then, a statistical decision is made based on the decision rule; i.e., accept or reject H0.
2. (5 points) Test the hypothesis that a population mean is 150 against the alternate that it is not 150. Assume a sample of 400 was taken and the sample mean was computed to be 152.2575 and the sample standard deviation was 15. Assume ratio level data. We do not know the population standard deviation.
Use formal hypothesis testing (the five step model) to do the test. Clearly state the conclusion both in statistical terms and in conversational English.
Step one: The population mean is 150.
The population mean is NOT 150.
Using the z-test and a 95% confidence level, the low of 150.7875 and the high of 153.727 leads us to accept the null hypothesis.
Compute the p-value
What are type I and type II errors? Which applies here and what are the consequences of the error?
Type I errors occur when we erroneously reject the null hypothesis. Sometimes, no matter how hard we try to accurately reflect the population parameters, false information can find its way into our testing. As a result, we can erroneously reject the null based on that false information. Type II errors are usually based on false assumptions. Type II errors typically occur in smaller sampling sizes. Our example is a Type I error.
3. (2 points) What hypothesis testing model would be used under each of the following conditions. The information given in each cell is: level of measurement, number of groups, sample size to be taken, parameter of interest and any qualifications.
Don't know sigma Generalized Likelihood Ratio
Populations are normally distributed Chi-square
Two factors of classification
Test for independence
Expected frequencies condition met Two-sample
Don't know sigma
Population is normally distributed Bayesian
4. (1 point) What is the purpose of statistics? (Not the cost issue.)
Statistics is the mathematical science that collects, analyzes, interprets and presents data. Statistics are valuable to the sciences and humanities on an academic level and are frequently used by both business and government. Statistics can be used to summarize or describe data (descriptive statistics), establish patterns (modeled statistics), or to make inferences about the population as a whole (inferential statistics). For example, the government uses statistics when it takes a population census. Businesses can use statistics to determine customer satisfaction. Statistics can be used in many areas, including medicine. When doctors or other health practitioners use control groups to test medical procedures or medications, they are using statistics.
5. (2 points) Why does the p-value approach to hypothesis testing "work"?
The p-value can cast doubt on the null hypothesis and drives us to undertake the full test to ensure that our conclusion is correct.
State the decision given the p values
Pvalue Alpha Decision
0.04 0.01. Chi-square
0.04 0.05 Bayesian
0.12 0.15 Randomized
This posting shows steps in formal hypothesis testing process.