Hypothesis Testing When you are evaluating a hypothesis, you need to account for both the variability in your sample and how large your sample is. Introduction Hypothesis testing is generally used when you are comparing two or more groups. Examples: There is no difference in intubation rates across ages 0 to 5 years.
The intervention and control groups have the same survival rate or, the intervention does not improve survival rate. There is no association between injury type and whether or not the patient received an IV in the prehospital setting. Step 2: Specify the Alternative Hypothesis The alternative hypothesis H 1 is the statement that there is an effect or difference. Examples: The intubation success rate differs with the age of the patient being treated two-sided. The time to resuscitation from cardiac arrest is lower for the intervention group than for the control one-sided.
There is an association between injury type and whether or not the patient received an IV in the prehospital setting two sided. Step 3: Set the Significance Level a The significance level denoted by the Greek letter alpha— a is generally set at 0. Step 4: Calculate the Test Statistic and Corresponding P-Value In another section we present some basic test statistics to evaluate a hypothesis.
Not likely to happen strictly by chance. Very likely to occur strictly by chance. Example: Average ages were significantly different between the two groups Is this an important difference? Probably not, but the large sample size has resulted in a small p-value. Example: Average ages were not significantly different between the two groups It could be, but because the sample size is small, we can't determine for sure if this is a true difference or just happened due to the natural variability in age within these two groups.
Your result is statistically significant. Your result is not statistically significant. After all, we took a random sample and our sample mean of That is different, right? Sampling error is the difference between a sample and the entire population. A hypothesis test helps assess the likelihood of this possibility!
In fact, if we took multiple random samples of the same size from the same population, we could plot a distribution of the sample means. A sampling distribution is the distribution of a statistic, such as the mean, that is obtained by repeatedly drawing a large number of samples from a specific population.
This distribution allows you to determine the probability of obtaining the sample statistic. Fortunately, I can create a plot of sample means without collecting many different random samples!
Our goal is to determine whether our sample mean is significantly different from the null hypothesis mean. The graph below shows the expected distribution of sample means. However, there is a reasonable probability of obtaining a sample mean that ranges from to , and even beyond! Kevin writes his hypotheses, remembering that Foothill will be making a decision about spending a fair amount of money based on what he finds.
When writing his hypotheses, Kevin knows that if his sample has a proportion of decorated socks well below. He only wants to say the data support the alternative if the sample proportion is well above. To include the low values in the null hypothesis and only the high values in the alternative, he uses a one-tail test, judging that the data support the alternative only if his z-score is in the upper tail.
He will conclude that the machinery should be bought only if his z-statistic is too large to have easily come from the sampling distribution drawn from a population with a proportion of. Kevin will accept H a only if his z is large and positive. Checking the bottom line of the t-table, Kevin sees that. If his sample z is greater than Using the data the salespeople collected, Kevin finds the proportion of the sample that is decorated:.
Figure 4. Because his sample calculated z-score is larger than John can feel comfortable making the decision to buy the embroidery and sewing machinery. We also use hypothesis testing when we deal with categorical variables. Categorical variables are associated with categorical data.
For instance, gender is a categorical variable as it can be classified into two or more categories. In business, and predominantly in marketing, we want to determine on which factor s customers base their preference for one type of product over others.
If it does, she will explore the idea of charging different prices for dishes popular with different age groups. The sales manager has collected data on sales of different dishes over the last six months, along with the approximate age of the customers, and divided the customers into three categories. Table 4. The underlying test for this contingency table is known as the chi-square test.
Then we calculate the expected frequency for the above table with i rows and j columns, using the following formula:. This chi-square distribution will have i -1 j -1 degrees of freedom. One technical condition for this test is that the value for each of the cells must not be less than 5. The expected frequency, E ij , is found by multiplying the relative frequency of each row and column, and then dividing this amount by the total sample size.
For each of the expected frequencies, we select the associated total row from each of the age groups, and multiply it by the total of the same column, then divide it by the total sample size. Now we use the calculated expected frequencies and the observed frequencies to compute the chi-square test statistic:.
We computed the sample test statistic as To find out the exact cut-off point from the chi-square table, you can enter the alpha level of. This template contains two sheets; it will plot the chi-square distribution for this example and will automatically show the exact cut-off point. The result indicates that our sample data supported the alternative hypothesis. Based on this outcome, the owner may differentiate price based on these different age groups.
Using the test of independence, the owner may also go further to find out if such dependency exists among any other pairs of categorical data. This time, she may want to collect data for the selected age groups at different locations of her restaurant in British Columbia. The results of this test will reveal more information about the types of customers these restaurants attract at different locations. Depending on the availability of data, such statistical analysis can also be carried out to help determine an improved pricing policy for different groups in different locations, at different times of day, or on different days of the week.
Finally, the owner may also redo this analysis by including other characteristics of these customers, such as education, gender, etc. This chapter has been an introduction to hypothesis testing. You should be able to see the relationship between the mathematics and strategies of hypothesis testing and the mathematics and strategies of interval estimation.
When making an interval estimate, you construct an interval around your sample statistic based on a known sampling distribution. When testing a hypothesis, you construct an interval around a hypothesized population parameter, using a known sampling distribution to determine the width of that interval. You then see if your sample statistic falls within that interval to decide if your sample probably came from a population with that hypothesized population parameter.
Hypothesis testing also has implications for decision-making in marketing, as we saw when we extended our discussion to include the test of independence for categorical data. Hypothesis testing is a widely used statistical technique. It forces you to think ahead about what you might find. By forcing you to think ahead, it often helps with decision-making by forcing you to think about what goes into your decision.
All of statistics requires clear thinking, and clear thinking generally makes better decisions. Hypothesis testing requires very clear thinking and often leads to better decision-making. Tiemann is licensed under a Creative Commons Attribution 4. Skip to content Main Body. Previous: Chapter 3. Making Estimates.
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