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-: Hi again.
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As you probably expected, in this lesson
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we will learn about independent samples with known variants.
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Let's get into the example right away.
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You may remember this one.
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We are about to test the average grades of students
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from two different departments in a UK university.
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I would like to remind you
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that in the UK grades are expressed in percentages.
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The two departments are engineering and management.
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We were told by the dean
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that engineering is a tougher discipline
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and people tend to get lower grades.
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He believes that on average management students
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outperform engineering students by four percentage points.
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Now it is our job to verify if that is the case.
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Let's state the two hypotheses.
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H zero is the difference
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between the means of the two populations is minus four.
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By the way, notice that I can make H zero engineering
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minus management and get a negative difference
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or I can make H zero management
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minus engineer and get a positive difference.
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Either way works.
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Just so we can see as many different situations as possible.
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I will keep the difference negative.
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So, H one is the population mean difference is different
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than four.
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Once again, this is a two-sided test.
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Our research question is not to find the difference
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but to check if it is exactly four.
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Right. Let's get our hands dirty.
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Here's the table that summarizes the data.
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The sample sizes are 170 respectively.
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The sample means our 58% and 65%
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and the population's standard deviations are 10%
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and 6% and are known from past data.
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If you remember, when the population is known
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for independent samples, the standard error
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of the difference is equal to the square root
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of the sum of the variance of engineering divided
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by its sample size and the variance of management,
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again divided by its sample size.
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All we have left is to compute the test statistic.
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We have big samples and known variances.
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Therefore, we can use the Z statistic.
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I hope you are getting the point.
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Small samples and unknown variances means T large sample
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and known variances mean Z.
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When we have large samples and unknown variances
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it is up to the researcher
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but generally it is okay to use Z in that case as well.
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All right, here's the formula for the test statistic.
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Sample difference mean
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minus hypothesized difference mean divided
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by the standard error.
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We plug in the numbers and get a Z score of minus 2.44.
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Let's calculate the P value.
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Once again, I'll just tell you the P value
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as usually you will obtain it using a software.
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The P value of the two-sided test is 0.015.
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What we can say is that at 5% significance, which is common
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for such a study, the P value of 0.015 is lower than 0.05.
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Thus, we reject the null hypothesis.
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There is enough statistical evidence
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that the difference of the two means is not 4%.
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All right, cool.
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Here's a trick.
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What if you wanna know
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if the difference is higher or lower than four?
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The sign of the test statistic
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can give you that information.
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A minus sign of the test statistic means it's smaller.
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If you reverse engineer the standardization process
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you'll find that true value is likely to be lower
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than the hypothesized value.
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In our case, this translates
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into the true mean is likely to be lower than minus four.
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Lower than minus four entails
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that possible values are minus five, minus six, and so on.
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This is additional information that you can give
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to the dean.
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All right, done with that lesson too
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let's proceed to the final topic.
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Independent samples and unknown variances.
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