Lab 14 Hypothesis testing

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Illinois State University *

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138

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Statistics

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Apr 3, 2024

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Lab 14: Hypothesis testing Step 1: Make a hypothesis - the assumptions are related to the stat test you'll do and we'll talk more about those as we discuss each individual test - your hypothesis is an educated guess/prediction about the effect of particular events/treatments/factors (which result in differences between populations) - your hypothesis may be general (e.g., this course will change comprehension abilities), or specific (e.g., this course will improve comprehension abilities by at least 10%). The standard logic that underlies hypothesis testing is that there are always (at least) two hypotheses: the null hypothesis and the alternative hypothesis The null hypothesis (H 0 ) predicts that the independent variable (treatment) has no effect on the dependent variable for the population. The alternative hypothesis (H a ) predicts that the independent variable will have an effect on the dependent variable for the population The hypothesis testing procedure assumes we are trying to reject the null hypothesis, not trying to prove the alternative hypothesis . Example : Suppose that we know that in the US on average 30% of registered voters vote in each election. You want to try to increase that number with an ad campaign to try to get more people to vote. So we conduct the ad campaign before a major election and then record the percentage of voters that vote in that election. What will our hypotheses be in this case? H 0 states that the independent variable will have no effect so our H 0 is that m = 30% (indicating no effect of ad campaign). Our H 1 is the opposite: that m will not equal 30%. Alternatively, we could make a specific alternative hypothesis if we chose. This would change our H 0 too. Let's consider the specific case above where we expect that the ad campaign will INCREASE voters. This means that we expect higher voting rates for our sample than is in the population (30%). Here our H a is that m > 30% . That means that our H 0 is m < or = 30%. Try some on your own. Each of the following situations calls for a significance test for a population mean m. State the null hypothesis H 0 and the alternative hypothesis H a in each case. (1a) Census Bureau data show that the mean household income in the area served by a shopping mall is $52,500 per year. A market research firm questions shoppers at the mall.

The researchers suspect the mean household income of mall shoppers is higher than that of the general population. H 0 : __the mean household income of mall shoppers is not higher than that of the general population_(<$52,500 or =$52,500)_____ H a : __the mean household income of mall shoppers is higher than that of the general population__(>$52,500)___ (1b) The examinations in a large psychology class are scaled after grading so that the mean score is 50. The professor thinks that one teaching assistant is a poor teacher and suspects that his students have a lower mean than the class as a whole. The TA's students this semester can be considered a sample from the population of all students in the course, so the professor compares their mean score with 50. H 0 : __the students with the poor TA will not have a lower mean than the class as a whole_(mean score = 50 or >50) H a : __the students with the poor TA will have a lower mean than the class as a whole (<50)____________________________________________________ Part 2: The Distribution of Sample Means Click on this link à Dist. of Sample Means and click Begin. The top distribution is the population distribution. We'll start a normally distributed population, which is set as the default. On the left are the descriptive statistics that describe this distribution. From this population distribution we can draw random samples.

(2a) Click on the "5" button below the “Animated” button. This will randomly select 5 samples (each sample will have n=5). The distribution of sample means plot (the third one down) will now have 5 sample means in it. What is the mean of the distribution of sample means (the mean of the 5 sample means)? __18.41________________ (2b) What is the standard deviation for the distribution of sample means? __1.23_________ (2c) How does this mean of the distribution of sample means compare to the actual population mean? _ the mean of the actual population is 16.00 à the mean from 2a is slightly higher ________________ Click the “clear lower 3” button. (3a) Now click the "100,000" button. This will take 100,000 samples (of size n=5) and plot all of the sample means on the distribution of sample means plot. What is the mean of the distribution of sample means? ____16.01_______________ (3b) What is the standard deviation for the distribution of sample means? ___2.24________ (3c) How does this mean of the distribution of sample means compare to the actual population mean? _ the mean of the actual population is 16.00 à the mean from 3a is the smallest bit different – they are very close_________ _______________________________ Standard Error The standard deviation of the distribution of sample means is called the standard error . The standard error is influenced by two factors: the variability of the population (s) and the sample size (n) . We'll consider each of these factors below: (A) the variability of the population - the bigger the variability of the population, the more variability you'll have in the sample means. large s big differences from the pop mean small s

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small differences from the pop mean (B) the size of the sample - the larger your sample size (n), the more accurately the sample represents the population. This is known as the Law of large numbers . think of it this way: - If I randomly selected 1 score, how accurately will that score predict the population's mean? - Suppose that I take 5 scores. Are things more accurate? - What about 100 scores? These two characteristics are combined in the formula for the standard error . standard error(SE) = ! √# (4a) Using the information from (2a) and (2b), what is the standard error when your sample size was 5? ______.55_______________ (4b) Using the information from (3a) and (3b), what is the standard error when your sample size was 5? ____1.00_________________

Lab 14 Hypothesis testing

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