Principles of Hypothesis
So far we have talked about estimating a confidence interval along with the probability (the confidence level) that the true population statistic lies within this interval under repeated sampling. We now examine the principles of statistical inference to hypotheses testing. By the end of this chapter you should be able to • Understand what is hypothesis testing • Examine issues relating to the determination of level of How is this Done?
If the difference between our hypothesized value and the sample value is small, then it is more likely that our hypothesized value of the mean is correct. The larger the difference the smaller the probability that the hypothesized value is correct. In practice however very rarely is the difference between the sample mean and the hypothesized population value larger enough or small enough for us to be able to accept or reject the hypothesis prima-facie. We cannot accept or reject a hypothesis about a parameter simply on intuition; instead we need to use objective criteria based on sampling theory to accept or reject the hypothesis.
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Hypotheses testing is the process of making inferences about a population based on a sample. The key question therefore in hypotheses testing is: how likely is it that a population such as one we have hypothesized to produce a sample such as the one we are looking at. significance • Apply tests of hypotheses to large to management Situations • Use of SPSS package to carry out hypotheses test and interpretation of computer output including p- values What is Hypothesis Testing? What is a Hypothesis? A hypothesis is the assumption that we make about the population parameter.
This can be any assumption about a population parameter not necessarily based on statistical data. For example it can also be based on the gut feel of a manager. Managerial hypotheses are based on intuition; the market place decides whether the manager’s intuitions were in fact correct. In fact managers propose and test hypotheses all the time. For example: • If a manager says ‘if we drop the price of this car model by Hypotheses Testing-The theory Null Hypothesis In testing our hypotheses we must state the assumed or hypothesized value of the population parameter before we begin sampling.
The assumption we wish to test is called the Null Hypotheses and is symbolized by Ho. For example if we want to test the hypotheses that the population mean is 500. We would write it as: Ho: µ=500 If we use the hypothesized value of a population mean in a problem we represent it symbolically as: µHo. The term null hypotheses has its origins in pharmaceutical testing where the null hypotheses is that the drug has no effect, i. e. , there is no difference between a sample treated with the drug and untreated samples. Alternative Hypothesis If our sample results fail to support the hypotheses we must conclude that something else must be true.
Whenever we reject the null hypothesis the alternative hypothesis is the one we have to accept. This symbolized by Ha . There are three possible alternative hypotheses for any Ho. , i. e. : Ha: µ? 500(the alternative hypothesis is not equal to 500) Ha: µ>500(the alternative hypothesis is greater than 500) Ha: µ<500( the alternative hypothesis is less than 500) Understanding Level of Significance The purpose of testing a hypothesis is not to question the computed value of the sample statistics but to make a judgment about the difference between the sample statistic and the hypothesized population parameter.
Therefore the next step, after stating our null and alternative hypotheses, is to decide what Rs15000 , we’ll increase sales by 25000 units’ is a hypothesis. To test it in reality we have to wait to the end of the year to and count sales. • A manager estimates that sales per territory will grow on average by 30% in the next quarter is also an assumption or hypotheses. How would the manager go about testing this assumption? Suppose he has 70 territories under him. • One option for him is to audit the results of all 70 territories and determine whether the average is growth is greater than or less than 30%.
This is a time consuming and expensive procedure. • Another way is to take a sample of territories and audit sales results for them. Once we have our sales growth figure, it is likely that it will differ somewhat from our assumed rate. For example we may get a sample rate of 27%. The manager is then faced with the problem of determining whether his assumption or hypothesized rate of growth of sales is correct or the sample rate of growth is more representative. To test the validity of our assumption about the population we collect sample data and determine the sample value of the statistic.
We then determine whether the sample data supports our hypotheses assumption regarding the average sales growth. 11. 556 © Copy Right: Rai University 113 criterion do we use for deciding whether to accept or reject the null hypothesis. How do We use Sampling to Accept or Reject Hypothesis? The Process of Hypothesis Testing We now look at the process of hypothesis testing. An example will help clarify the issues involved: Aluminum sheets have to have an average thickness of . 04inches or they are useless.
A contractor takes a sample of 100 sheets and determines mean sample thickness as . 0408 inches. On the basis of past experience he knows that the population standard deviation for these sheets is . 04 inches. The issue the contractor faces is whether he should , on the basis of sample evidence, accept or reject a batch of 10,000 aluminum sheets. In terms of hypotheses testing the issue is : • If the true mean is . 04inches and the standard deviation.
We use the result that there is a certain fixed probability associated with intervals from the mean defined in terms of number of standard deviations from the mean. Therefore our problem of testing a hypothesis reduces to determining the probability that a sample statistic such as the one we have obtained could have arisen from a population with a hypothesized mean m. In the hypothesis tests we need two numbers to make our decision whether to accept or reject the null hypothesis: • an observed value or computed from the sample • a critical value defining the boundary between the acceptance and rejection region .
Instead of measuring the variables in original units we calculate a standardized z variable for a standard normal distribution with mean µ=0. The z statistic tells us how many how many standard deviations above or below the mean standardized mean (z,<0, z>0) our observation falls. We can convert our observed data into the standardized scale using the transformation .004 inches, what are the chances of getting a sample mean that differs from the population mean (. 04 inches) by . 0008inches or more? To find this out we need to calculate the probability that a random sample with mean . 08 will be selected from a population with µ =. 04 and a standard deviation.
If this probability is too low we must conclude that the aluminum company’s statement is false and the mean thickness of the consignment supplied is not . 04inches. Once we have stated out hypothesis we have to decide on a criterion to be used to accept or reject Ho. The level of significance represents the criterion used by the decision maker to accept or reject a hypothesis. For example if the manager wishes to allow for a 5% level of significance.
This means that we reject the null hypothesis when the observed difference between the sample mean and population mean is such that it or a larger difference would only occur 5 or less times in every 100 samples when the hypothesized value of the population parameter is correct. It therefore indicates the permissible extent of sampling variation we are willing to allow whilst accepting the null hypothesis. In statistical terms 5% is called the level of significance and is denoted by a=. 05 We now write our data systematically.
The z statistic measures the number of standard deviations away from the hypothesized mean the sample mean lies. From the standard normal tables we can calculate the probability of the sample mean differing from the true population mean by a specified number of standard deviations. For example: • we can find the probability that the sample mean differs from the population mean by two or more standard deviations. It is this probability value that will tell us how likely it is that a given sample mean can be obtained from a population with a hypothesized mean m. . • If the probability is low for example less than 5% , perhaps
Our sample data is as follows: n=100, it can be reasonably concluded that the difference between the sample mean and hypothesized population mean is too large and the chance that the population would produce such a random sample is too low. What probability constitutes too low or acceptable level is a judgment for decision makers to make. Certain situations demand that decision makers be very sure about the characteristics of the items being tested and even a 2% probability that the population produces such a sample is too high.
In other situations there is greater latitude and a decision maker may be wiling to accept a hypothesis with a 5% probability of chance variation. In each situation what needs to be determined are the costs resulting from an incorrect decision and the exact level of risk we are willing to assume. Our minimum standard for an acceptable probability, say, 5%, is also the risk we run of rejecting a hypothesis that is true. To test any hypothesis we need to calculate the standard error of the mean from the population standard deviation
Next we calculate the z statistic to determine how many standard errors away from the true mean our sample mean is. This gives us our observed value of z which can then be compared with the z critical from the normal tables. z= = . 0408-. 04/. 0004=2 x ? µ ? x This is demonstrated in the figure 1 below. 114 © Copy Right: Rai University 11. 556 Interpreting the Level of Significance The level of significance is demonstrated diagrammatically below in figure 3. Here . 95 of the area under the curve is where we would accept the null hypotheses.