Where is hypothesis testing used

Statisticians know the sampling distribution of a statistic that compares the expected frequency of a sample with the actual, or observed , frequency. The expected frequency, E, is found by multiplying the relative frequency of this class in the H o hypothesized population by the sample size.

This gives you the number in that class in the sample if the relative frequency distribution across the classes in the sample exactly matches the distribution in the population. David now needs to come up with a rule to decide if the data support H o or H a. He looks at the table and sees that for 5 df there are 6 classes—there is an expected frequency for size 11 socks , only.

He decides that it would not be all that surprising if the players had a different distribution of sock sizes than the athletes who are currently buying Easy Bounce, since all of the players are women and many of the current customers are men. As a result, he uses the smaller. He starts by finding the expected frequency of size 6 socks by multiplying the relative frequency of size 6 in the population being produced by 97, the sample size.

He then realizes that he will have to do the same computation for the other five sizes, and quickly decides that a spreadsheet will make this much easier see Table 4. David performs his third step, computing his sample statistic, using the spreadsheet. David has found that his sample data support the hypothesis that the distribution of sock sizes of the players is different from the distribution of sock sizes that are currently being manufactured. If Easy Bounce socks are successfully marketed to the BC college players, the mix of sizes manufactured will have to be altered.

Now review what David has done to test to see if the data in his sample support the hypothesis that the world is unsurprising and that the players have the same distribution of sock sizes as the manufacturer is currently producing for other athletes.

Formally, David first wrote null and alternative hypotheses, describing the population his sample comes from in two different cases. The first case is the null hypothesis; this occurs if the players wear socks of the same sizes in the same proportions as the company is currently producing.

The second case is the alternative hypothesis; this occurs if the players wear different sizes. After he wrote his hypotheses, he found that there was a sampling distribution that statisticians knew about that would help him choose between them.

Acting on this finding, David will include a different mix of sizes in the sample packages he sends to team coaches. As you learned in Chapter 3 , sample proportions can be used to compute a statistic that has a known sampling distribution. Reviewing, the z-statistic is:. If you look at the z-table, you can see that. The basic strategy is the same as that explained earlier in this chapter and followed in the goodness-of-fit example: a write two hypotheses, b find a sample statistic and sampling distribution that will let you develop a decision rule for choosing between the two hypotheses, and c compute your sample statistic and choose the hypothesis supported by the data.

Foothill did not have the right machinery to sew on the embroidered patches and contracted out the sewing. 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. Since this value is above. Hypothesis testing When interpreting research findings, researchers need to assess whether these findings may have occurred by chance. Basic concepts Null and research hypothesis Probability value and types of errors Effect size and statistical significance Directional and non-directional hypotheses Null and research hypotheses To carry out statistical hypothesis testing, research and null hypothesis are employed: Research hypothesis : this is the hypothesis that you propose, also known as the alternative hypothesis HA.

For example: H A: There is a relationship between intelligence and academic results. H A; Males and females differ in their levels of stress. The null hypothesis H o is the opposite of the research hypothesis and expresses that there is no relationship between variables, or no differences between groups; for example: H o : There is no relationship between intelligence and academic results.

H o : Males and females will not differ in their levels of stress. In hypothesis testing, a value is set to assess whether the null hypothesis is accepted or rejected and whether the result is statistically significant: A critical value is the score the sample would need to decide against the null hypothesis.

A probability value is used to assess the significance of the statistical test. If the null hypothesis is rejected, then the alternative to the null hypothesis is accepted. Probability value and types of errors The probability value, or p value , is the probability of an outcome or research result given the hypothesis.

There are two types of errors associated to hypothesis testing: What if we observe a difference — but none exists in the population? What if we do not find a difference — but it does exist in the population? These situations are known as Type I and Type II errors: Type I Error: is the type of error that involves the rejection of a null hypothesis that is actually true i. Type II Error: is the type of error that occurs when we do not reject a null hypothesis that is false i.

Effect size and statistical significance When the null hypothesis is rejected, the effect is said to be statistically significant. A directional hypothesis specifies the direction of the relationship between independent and dependent variables. A hypothesis that states that students who attend an intensive Statistics course will obtain higher grades than students who do not attend would be directional.

A non-directional hypothesis states that there will be a difference, but we do not know what direction that will be. The hypothesis only states that they will obtain different grades. The hypothesis testing process The hypothesis testing process can be divided into five steps: Restate the research question as research hypothesis and a null hypothesis about the populations. Determine the characteristics of the comparison distribution. Determine the cut off sample score on the comparison distribution at which the null hypothesis should be rejected.

Decide whether to reject the null hypothesis. The experiment involves two groups of students: the first group consumes caffeine; the second group drinks water.

Both groups complete a memory test. A randomly selected individual in the experimental condition i. The scores of people in general on this memory measure are normally distributed with a mean of 19 and a standard deviation of 4.

The researcher predicts an effect differences in memory for these groups but does not predict a particular direction of effect i. Step 1 : There are two populations of interest. The economist randomly samples 25 families and records their energy costs for the current year.

Read on! Why do we even need hypothesis tests? 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. The smaller the significance level, the greater the burden of proof needed to reject the null hypothesis, or in other words, to support the alternative hypothesis.

In another section we present some basic test statistics to evaluate a hypothesis. Hypothesis testing generally uses a test statistic that compares groups or examines associations between variables. When describing a single sample without establishing relationships between variables, a confidence interval is commonly used. The p-value describes the probability of obtaining a sample statistic as or more extreme by chance alone if your null hypothesis is true.

This p-value is determined based on the result of your test statistic. Your conclusions about the hypothesis are based on your p-value and your significance level.

Cautions About P-Values Your sample size directly impacts your p-value. Large sample sizes produce small p-values even when differences between groups are not meaningful. You should always verify the practical relevance of your results. On the other hand, a sample size that is too small can result in a failure to identify a difference when one truly exists. Plan your sample size ahead of time so that you have enough information from your sample to show a meaningful relationship or difference if one exists.

See calculating a sample size for more information. If you do a large number of tests to evaluate a hypothesis called multiple testing , then you need to control for this in your designation of the significance level or calculation of the p-value. For example, if three outcomes measure the effectiveness of a drug or other intervention, you will have to adjust for these three analyses.