Hypothesis Testing

Hypothesis testing is basically a process that uses statistical inference to test claims about population parameters. In hypothesis testing we begin by making a tentative assumption about a population parameter. This assumption is called the null hypothesis and is denoted by Ho. We then define another hypothesis, called the alternative hypothesis, which is the opposite of what is stated is the null hypothesis. The alternative hypothesis is denoted by Ha. The hypothesis testing procedure involves using data from a sample to test the competing claims indicated by the null and alternate hypotheses.

Testing the hypothesis is similar to a criminal trail. Ho: The defendant is innocent Ha: The defendant is guilty
Just as the defendant in a criminal trial is assumed to be innocent, the null hypothesis is assumed to be true and given the benefit of the doubt. The testimony and evidence obtained during the trial provide the sample information. If the sample information is consistent with the assumption of innocence, the null hypothesis that the defendant is innocent cannot be rejected. On the other hand, if the sample information is inconsistent with the assumption of innocence, the null hypothesis is rejected and action is taken based on the alternative hypothesis that the defendant is guilty.

Developing Null and Alternative Hypotheses

In some applications it may not be obvious how the null and alternative hypotheses should be formulated. Care must be taken to ensure that the hypotheses are structured appropriately and that the hypothesis-testing provides the information that the decision maker desires. The following are some guidelines for formulating the hypotheses for three types of situations where hypothesis-testing is frequently employed.

Testing Research Hypotheses

Consider a case where a particular model automobile currently obtains an average of 25 miles per gallon of gas. A product research group has developed a new carburetor specifically designed to increase the miles per gallon performance. In order to evaluate the new design, several new carburetors will be manufactured, installed in automobiles, and subjected to research controlled driving tests. Notice that the product research group is looking for evidence to conclude that the new design increases the miles per gallon. In this case the research hypothesis is that the new carburetor will provide a mean miles per gallon exceeding 25; that is, [pic]. As a general guideline, a research hypothesis such as this is formulated as the alternative hypothesis. Thus appropriate null and alternative hypotheses for this study are as follows: Ho: [pic]< 25 Ha: [pic]> 25
If the sample results indicate that Ho cannot be rejected, we will not conclude that the carburetor improves mileage. Perhaps more research and testing needs to be conducted. If the sample data suggests that the null hypothesis can be rejected, the inference is made that [pic]> 25 since the researcher now has the necessary statistical support.
In research studies such as these, the null and alternate hypotheses should be formulated so that the rejection of the null, and hence the inference that the alternative is true, will provide the researcher with the conclusion and action being sought. The research hypothesis then should be expressed as the alternative hypothesis.

Testing Validity of Assumption

Testing Validity of Assumptions usually occur when you are testing the claims made by another entity. For example, a manufacturer of soft drinks states that their 2-liter bottle contains an average of 67 fluid ounces. Testing this will entail taking a sample of the 2-liter bottles and measuring their contents. In terms of the hypotheses formulation you give the company the benefit of the doubt and assume that their claim is true. Using this approach the hypothesis will be stated as given below: H0: [pic]> 67 Ha: [pic]< 67
In a situation where we are testing the validity of an assumption, the null hypothesis is generally formulated based on the assumption. The alternate hypothesis is formulated so that the rejection of the null will provide statistical evidence that the stated assumption is incorrect.

Decision Making

A hypothesis-testing situation involving decision making occurs when a decision maker must choose between two courses of action, one associated with the null hypothesis and the other with the alternate hypothesis. For example, on the basis of a sample of parts from a shipment that has just been received, a quality control inspector must decide whether to accept the shipment or return it to the supplier because it does not meet specifications. Assume that a specification for the part indicates a mean length of 1 inch per part is required. If the average length of the part is greater than or less than the 1inch standard, the parts will cause quality problems. In this case the null and alternative hypotheses would be formulated as follows: H0: [pic]= 1 Ha: [pic][pic]1

Testing the Hypotheses

Once the hypotheses have been formulated the next step, is to test the hypotheses. For clarity purposes I am listing the steps that we need to perform as follows: 1. In most cases the first step involves collecting data based on a sample and summarizing the data to obtain the sample mean and the standard deviation of the sample. 2. At this point we draw from our knowledge regarding sampling distributions. We know that for large sample cases (n > 30), the sampling distribution of sample means is normally distributed with the mean equal to the population mean and the dispersion measure equal to the standard error. In other words, if we kept taking samples of the same size, measured the mean for each sample, and charted the means of each sample, the resultant distribution will be normally distributed (i.e. bell shaped). The mean for the distribution will equal the mean for the population. This makes intuitive sense since the mean of all possible sample means should be very close to the population mean itself. 3. The standard deviation of the above distribution, referred to as the standard error is calculated as Sx = S/[pic] (where S is the standard deviation from the sample and Sx is the standard error). In cases where the population standard deviation is known, the above formula is still valid except that we substitute [pic](the population standard deviation) wherever we have the sample standard deviation. We need not belabor this point any further, take this for granted at present and bring any further discussions to class. 4. In a hypothesis test, the mean indicated on the null hypothesis is assumed to be true, and hence is the mean for the normal distribution of the sample means. The standard error for the distribution is calculated as in step 3. 5. Now if the null hypothesis is true, then the mean from any sample that we take will have a much higher probability of being closer to the hypothesized mean than the probability of being far away from the hypothesized mean. The process for the test is then to define a statistical limit, based on a confidence level, to how far away the sample mean can be from the hypothesized mean for us to conclude that it is close enough. Once these limits are defined we check if the sample statistic (mean) is within this limit. If it is within the limit we assume that the null is true. If it is outside the limits the alternative hypothesis is accepted.

Problems for Class

1. The National Optical Company presently uses A-1 polishing pads to polish eyeglass lenses. Pad life is critical, as National spends $15,000 per day on these pads. B-2 vendor claims that its pads have a longer polishing life. The currently used A-1 pads last, on average, 25 minutes before they must be replaced. Given that both vendors are noted for their quality and charge the same for their pads, National will switch only if the B-2 pads actually last more than 25 minutes. The decision National has to make is whether to switch vendors or not. As a test National used 200 B-2 pads. The sample mean number of minutes before failure was observed to be 24.7 minutes and the sample standard deviation was 3.5 minutes. Using a 0.05 level of significance what decision should National make?

Ho: M > 25 – ok One Tailed Test Ha: M < 25

Test 3.5/SQRT(200) = .25(standard error) N = 200 25+/- .41 (1.65*.25) X = 24.7 24.59 S.D = 3.5

2. A Smooth and Creamy Ice Cream Company, sells 32-ounce cartons of Fudge Delight ice cream. The production process is highly automated. The filling machine for the 32-ounce carton is good but not perfect. It can go out of adjustment and fill less or more than 32 ounces in a carton. Periodically, several samples of the 32-ounce carton are tested to determine the actual number of ounces in the carton. Based on the results the machine is either adjusted or allowed to continue. A recent line graph has indicated that the filling process has been stationary for the past six months. In today’s test of 36 32-ounce cartons, the sample mean number of ounces per carton was 31.67 ounces and the sample standard deviation was 0.66 ounces. Based on the sample evidence will you adjust the machine using a significance level of 0.05 ([pic]= 0.025)?

Ho: M=32 Two Tailed Test
Ha: M=/32 – ok (95% not right)

N = 36 standard error = .66/SQRT 36 = 0.11
X = 31.67 or 32 32+/-.215 (1.96*.11)
S.D. = 0.66 31.785 - 32.215
Alpha = .025

3. The EPA has ordered Chemdump Inc. to reduce the mean level of a toxin chemical in its wastewater to less than 20 parts per million (ppm). Chemdump which has a poor track record of meeting the EPA’s standard, has recently developed a new waste treatment process that it hopes will meet the standard. Below are the toxic chemical levels in 20 samples of the Chemdump’s wastewater taken by the EPA over a four week period since the new process was in place. What action should the EPA take? Use [pic]=0.03)

| |Day 1 |Day 2 |Day 3 |Day 4 |Day 5 |
|Week1 |19.5 |20.0 |18.5 |18.7 |19.5 |
|Week 2 |20.7 |18.5 |19.4 |20.5 |19.8 |
|Week 3 |19.1 |20.3 |20.1 |19.6 |20.3 |
|Week 4 |18.7 |19.5 |19.7 |19.1 |18.3 |

Mean = 19.49
Standard error = 0.03/SQRT(20) = .01
99% confident it’s under 20