Choosing the Sample Size for Estimating M

What constitutes reasonable certainty? In most situations, the confidence level is set at 95 or 90, partly because of tradition and partly because these lev- els represent to some people a reasonable level of certainty. The 95 or 90 level translates into a long-run chance of 1 in 20 or 1 in 10 of not covering the pop- ulation parameter. This seems reasonable and is comprehensible, whereas 1 chance in 1,000 or 1 in 10,000 is too small. The tolerable error depends heavily on the context of the problem, and only someone who is familiar with the situation can make a reasonable judgment about its magnitude. When considering a confidence interval for a population mean m, the plus-or- minus term of the confidence interval is . Three quantities determine the value of the plus-or-minus term: the desired confidence level which determines the z-value used, the standard deviation s, and the sample size. Usually, a guess must be made about the size of the population standard deviation. Sometimes an initial sample is taken to estimate the standard deviation; this estimate provides a basis for determining the additional sample size that is needed. For a given toler- able error, once the confidence level is specified and an estimate of s supplied, the required sample size can be calculated using the formula shown here. Suppose we want to estimate m using a 1001 ⫺ a confidence interval hav- ing tolerable error W. Our interval will be of the form where E ⫽ W 兾2. Note that W is the width of the confidence interval. To determine the sample size n, we solve the equation for n. This formula for n is shown here: E ⫽ z a 兾2 s 兾1n y ⫾ E, z a 兾2 s 兾1n Sample Size Required for a 1001 ⴚ A Confidence Interval for M of the Form y – ⴞ E n ⫽ z a 兾2 2 s 2 E 2 Note that determining a sample size to estimate m requires knowledge of the population variance s 2 or standard deviation s. We can obtain an approximate sample size by estimating s 2 , using one of these two methods: 1. Employ information from a prior experiment to calculate a sample vari- ance s 2 . This value is used to approximate s 2 . 2. Use information on the range of the observations to obtain an estimate of s. We would then substitute the estimated value of s 2 in the sample-size equation to determine an approximate sample size n. We illustrate the procedure for choosing a sample size with two examples. EXAMPLE 5.3 The relative cost of textbooks to other academic expenses has risen greatly over the past few years, university officials have started to include the average amount ex- pended on textbooks into their estimated yearly expenses for students. In order for these estimates to be useful, the estimated cost should be within 25 of the mean expenditure for all undergraduate students at the university. How many students should the university sample in order to be 95 confident that their estimated cost of textbooks will satisfy the stated level of accuracy? Solution From data collected in previous years, the university officials have deter- mined that the annual expenditure for textbooks has a histogram that is normal in shape with costs ranging from 250 to 750. An estimate of s is required to find the sample size. Because the distribution of book expenditures has a normal like shape, a reasonable estimate of s would be The various components in the sample size formula are level of accuracy ⫽ E ⫽ 25, , and level of confidence ⫽ 95 which implies z a 兾2 ⫽ z .052 ⫽ z .025 ⫽ 1.96. Substituting into the sample-size formula, we have To be on the safe side, we round this number up to the next integer. A sample size of 97 or larger is recommended to obtain an estimate of the mean textbook expen- diture that we are 95 confident is within 25 of the true mean. EXAMPLE 5.4 A federal agency has decided to investigate the advertised weight printed on car- tons of a certain brand of cereal. The company in question periodically samples cartons of cereal coming off the production line to check their weight. A summary of 1,500 of the weights made available to the agency indicates a mean weight of 11.80 ounces per carton and a standard deviation of .75 ounce. Use this informa- tion to determine the number of cereal cartons the federal agency must examine to estimate the average weight of cartons being produced now, using a 99 confidence interval of width .50. Solution The federal agency has specified that the width of the confidence inter- val is to be .50, so E ⫽ .25. Assuming that the weights made available to the agency by the company are accurate, we can take s ⫽ .75. The required sample size with z a 兾2 ⫽ 2.58 is Thus, the federal agency must obtain a random sample of 60 cereal cartons to estimate the mean weight to within ⫾.25.

5.4 A Statistical Test for M

A second type of inference-making procedure is statistical testing or hypothesis testing. As with estimation procedures, we will make an inference about a popu- lation parameter, but here the inference will be of a different sort. With point and interval estimates, there was no supposition about the actual value of the param- eter prior to collecting the data. Using sampled data from the population, we are simply attempting to determine the value of the parameter. In hypothesis testing, there is a preconceived idea about the value of the population parameter. For ex- ample, in studying the antipsychotic properties of an experimental compound, we might ask whether the average shock-avoidance response of rats treated with a specific dose of the compound is greater than 60, m ⬎ 60, the value that has been observed after extensive testing using a suitable standard drug. Thus, there are two theories or hypotheses involved in a statistical study. The first is the hypothesis n ⫽ 2.58 2 .75 2 .25 2 ⫽ 59.91 n ⫽ 1.96 2 125 2 25 2 ⫽ 96.04 ˆs ⫽ 125 ˆ s ⫽ range 4 ⫽ 750 ⫺ 250 4 ⫽ 125 being proposed by the person conducting the study, called the research hypothesis, m ⬎ 60 in our example. The second theory is the negation of this hypothesis, called the null hypothesis, m ⱕ 60 in our example. The goal of the study is to decide whether the data tend to support the research hypothesis. A statistical test is based on the concept of proof by contradiction and is composed of the five parts listed here. research hypothesis null hypothesis statistical test 1. Research hypothesis also called the alternative hypothesis, denoted by H a . 2. Null hypothesis, denoted by H .

3.

Test statistics, denoted by T.S. 4. Rejection region, denoted by R.R. 5. Check assumptions and draw conclusions. For example, the Texas AM agricultural extension service wants to deter- mine whether the mean yield per acre in bushels for a particular variety of soybeans has increased during the current year over the mean yield in the previous 2 years when m was 520 bushels per acre. The first step in setting up a statistical test is determining the proper specification of H and H a . The following guidelines will be helpful: 1. The statement that m equals a specific value will always be included in H . The particular value specified for m is called its null value and is denoted m . 2. The statement about m that the researcher is attempting to support or detect with the data from the study is the research hypothesis, H a .

3.

The negation of H a is the null hypothesis, H . 4. The null hypothesis is presumed correct unless there is overwhelming evidence in the data that the research hypothesis is supported. In our example, m is 520. The research statement is that yield in the current year has increased above 520; that is, H a : m ⬎ 520. Note that we will include 520 in the null hypothesis. Thus, the null hypothesis, the negation of H a , is H : m ⱕ 520. To evaluate the research hypothesis, we take the information in the sample data and attempt to determine whether the data support the research hypothesis or the null hypothesis, but we will give the benefit of the doubt to the null hypothesis. After stating the null and research hypotheses, we then obtain a random sample of 1-acre yields from farms throughout the state. The decision to state whether or not the data support the research hypothesis is based on a quantity computed from the sample data called the test statistic. If the population distribution is determined to be mound shaped, a logical choice as a test statistic for m is or some function of . If we select as the test statistic, we know that the sampling distribution of is approximately normal with a mean m and standard deviation provided the population distribution is normal or the sample size is fairly large. We are at- tempting to decide between H a : m ⬎ 520 or H : m ⱕ 520. The decision will be to either reject H or fail to reject H . In developing our decision rule, we will assume that m ⫽ 520, the null value of m. We will now determine the values of , called the rejection region, which we are very unlikely to observe if m ⫽ 520 or if m is any other value in H . The rejection region contains the values of that support the research hypothesis and contradict the null hypothesis, hence the region of values for that reject the null hypothesis. The rejection region will be the values of in the upper tail of the null distribution m ⫽ 520 of . See Figure 5.5. y y y y y s 兾1n, y y y y test statistic rejection region