STA 2023 Calculator Instructions: Estimation and Hypothesis Testing Two PopulationsConfidence Interval for µ1 − µ2If you are fortunate enough to have information about the population standard deviations, you can use 2-SampZInt to estimate the difference µ1 − µ2 between two population means.According to the U.S. Bureau of the Census, the average annual salary of fulltime state employees was $49,056 in New York and $46,800 in Massachusetts in 2001. Suppose that these mean salaries are based on random samples of 500 full-time state employees from New York and 400 full-time state employees from Massachusetts and that the population standard deviations of the 2001 salaries of all full-time state employees in these two states were $9000 and $8500, respectively. Construct a 95% confidence interval for the difference in 2001 mean salaries between the two states.STA 2023 Calculator Instructions: Estimation and Hypothesis Testing Two PopulationsHypothesis Testing: µ1 = µ2If you are fortunate enough to have information about the population standard deviations, you can use 2-SampZTest to test whether or not µ1 = µ2 for two population means.STA 2023 Calculator Instructions: Estimation and Hypothesis Testing Two PopulationsPaired SamplesThe confidence intervals and hypothesis tests described above all assume that the samples are taken independently. Two samples that are taken from the same population are said to be dependent. The most common data collection design with dependent samples is called Pretest/Posttest. Data is collected from a sample before some type of treatment and then data is collected again from that same sample after the treatment. We then work with the mean difference between the pre and posttest scores. The null hypothesis is that the average difference is zero. By working with the differences between the variables, we can perform a one-sample T-Test.Example: Blood PressureSTA 2023 Calculator Instructions: Estimation and Hypothesis Testing Two PopulationsSTA 2023 Calculator Instructions: Estimation and Hypothesis Testing Two PopulationsTwo Population ProportionsWhen estimating the difference of two population proportions, p1−p2, or testing to see if they are equal, we can use the functions 2-PropZInt or 2-PropZTest, which are analogous to 1-PropZInt and 1-PropZTest. The functions are also located on the STAT page in the TEST list. As before, in order to use Z-based confidence intervals and tests for two populations we need sample sizes to be at least 30.Example: ToothpasteSTA 2023 Calculator Instructions: Estimation and Hypothesis Testing Two PopulationsConfidence Interval for µ1 − µ2If you are fortunate enough to have information about the population standard deviations, youcan use 2-SampZInt to estimate the difference µ1 − µ2 between two population means.The most general approach, though, to estimating the difference of two population means is tocompute a T-based confidence interval, which does not make any assumptions about thepopulation’s standard deviations. This is provided on the calculator by the function 2-SampTInt.Both of the interval functions are located on the STAT page in the TESTS list.Example: Average SalariesAccording to the U.S. Bureau of the Census, the average annual salary of fulltime stateemployees was $49,056 in New York and $46,800 in Massachusetts in 2001. Suppose that thesemean salaries are based on random samples of 500 full-time state employees from New York and400 full-time state employees from Massachusetts and that the population standard deviations ofthe 2001 salaries of all full-time state employees in these two states were $9000 and $8500,respectively. Construct a 95% confidence interval for the difference in 2001 mean salariesbetween the two states.Let New York employees be Population no.1 and Massachusetts employees be Population no.2.We are given the population standard deviations, so we will use 2-SampZInt with Stats selected.We set σ1 = 9000, σ2 = 8500, X1 = 49,056, n1 = 500, X2 = 46,800, n2 = 400, and our C-Level at0.95. Press the STAT key.Press the ► key twice to highlight TESTS.Press the number 9 key.Move the cursor over Stats and press the ENTER key.Type in 9000 for σ1.Type in 8500 for σ2.Type in 49056 for X1.Type in 500 for n1.Type in 46800 for X2.Type in 400 for n2.Type in .95 for the C-Level.1STA 2023 Calculator Instructions: Estimation and Hypothesis Testing Two PopulationsMove the cursor over Calculate and press the ENTER key.The 2-SampZInt output shows the: confidence interval: (1108.8, 3403.2) sample means and sample sizes.The 95% confidence interval for µ1 − µ2 is ($1108.8, $3403.2).Hypothesis Testing: µ1 = µ2If you are fortunate enough to have information about the population standard deviations, youcan use 2-SampZTest to test whether or not µ1 = µ2 for two population means.The more general approach to performing a hypothesis test on two population means is toperform a T-based test, which does not make any assumptions about the population’s standarddeviations. This is provided on the calculator by the function 2-SampTTest. Both of thesefunctions are located on the STAT page in the TESTS list.If you have the actual data stored in lists select Data and enter the names of the lists. If insteadyou have the summary statistics, select Stats. If you have reason to believe that the populationstandard deviations are the same, select Yes by the prompt Pooled.Example: Average SalariesTest at the 1% significance level if the 2001 mean salaries of full-time state employees in NewYork and Massachusetts are different.We are testing to choose between H0: µ1 = µ2 and H1: µ1 ø µ2. Since we have the populationstandard deviations, we will use 2-SampZTest.Press the STAT key.Press the ► key twice to highlight TESTS.Press the number 3 key.2STA 2023 Calculator Instructions: Estimation and Hypothesis Testing Two PopulationsMove the cursor over Stats and press the ENTER key.Type in 9000 for σ1.Type in 8500 for σ2.Type in 49056 for X1.Type in 500 for n1.Type in 46800 for X2.Type in 400 for n2.Move the cursor over ≠μ2 and press the ENTER key.Move the cursor over Calculate and press the ENTER key.The 2-SampZTest output shows the: alternative hypothesis: μ1≠ μ2 test statistic: Z=3.854158502 p-value: P=1.1617127E-4 sample means and sample sizes The p-value is