difference between two population means

The following data summarizes the sample statistics for hourly wages for men and women. When considering the sample mean, there were two parameters we had to consider, \(\mu\) the population mean, and \(\sigma\) the population standard deviation. And \(t^*\) follows a t-distribution with degrees of freedom equal to \(df=n_1+n_2-2\). The differences of the paired follow a normal distribution, For the zinc concentration problem, if you do not recognize the paired structure, but mistakenly use the 2-sample. The null and alternative hypotheses will always be expressed in terms of the difference of the two population means. There were important differences, for which we could not correct, in the baseline characteristics of the two populations indicative of a greater degree of insulin resistance in the Caucasian population . We can thus proceed with the pooled t-test. Therefore, $$ { t }_{ { n }_{ 1 }+{ n }_{ 2 }-2 }=\frac { { \bar { x } }_{ 1 }-{ \bar { x } }_{ 2 } }{ { S }_{ p }\sqrt { \left( \frac { 1 }{ { n }_{ 1 } } +\frac { 1 }{ { n }_{ 2 } } \right) } } $$. A difference between the two samples depends on both the means and the standard deviations. The \(99\%\) confidence level means that \(\alpha =1-0.99=0.01\) so that \(z_{\alpha /2}=z_{0.005}\). The null hypothesis will be rejected if the difference between sample means is too big or if it is too small. Describe how to design a study involving Answer: Allow all the subjects to rate both Coke and Pepsi. Another way to look at differences between populations is to measure genetic differences rather than physical differences between groups. Therefore, we do not have sufficient evidence to reject the H0 at 5% significance. We assume that \(\sigma_1^2 = \sigma_1^2 = \sigma^2\). Null hypothesis: 1 - 2 = 0. Monetary and Nonmonetary Benefits Affecting the Value and Price of a Forward Contract, Concepts of Arbitrage, Replication and Risk Neutrality, Subscribe to our newsletter and keep up with the latest and greatest tips for success. The same process for the hypothesis test for one mean can be applied. Further, GARP is not responsible for any fees or costs paid by the user to AnalystPrep, nor is GARP responsible for any fees or costs of any person or entity providing any services to AnalystPrep. Our test statistic lies within these limits (non-rejection region). Using the p-value to draw a conclusion about our example: Reject\(H_0\) and conclude that bottom zinc concentration is higher than surface zinc concentration. We want to compare the gas mileage of two brands of gasoline. In particular, still if one sample can of size \(30\) alternatively more, if the other is of size get when \(30\) the formulas of this section have be used. Carry out a 5% test to determine if the patients on the special diet have a lower weight. To perform a separate variance 2-sample, t-procedure use the same commands as for the pooled procedure EXCEPT we do NOT check box for 'Use Equal Variances.'. Denote the sample standard deviation of the differences as \(s_d\). Basic situation: two independent random samples of sizes n1 and n2, means X1 and X2, and Unknown variances \(\sigma_1^2\) and \(\sigma_1^2\) respectively. What were the means and median systolic blood pressure of the healthy and diseased population? Here "large" means that the population is at least 20 times larger than the size of the sample. For a right-tailed test, the rejection region is \(t^*>1.8331\). The formula for estimation is: Now let's consider the hypothesis test for the mean differences with pooled variances. From an international perspective, the difference in US median and mean wealth per adult is over 600%. Note that these hypotheses constitute a two-tailed test. [latex]\begin{array}{l}(\mathrm{sample}\text{}\mathrm{statistic})\text{}±\text{}(\mathrm{margin}\text{}\mathrm{of}\text{}\mathrm{error})\\ (\mathrm{sample}\text{}\mathrm{statistic})\text{}±\text{}(\mathrm{critical}\text{}\mathrm{T-value})(\mathrm{standard}\text{}\mathrm{error})\end{array}[/latex]. This test apply when you have two-independent samples, and the population standard deviations \sigma_1 1 and \sigma_2 2 and not known. The first step is to state the null hypothesis and an alternative hypothesis. We have \(n_1\lt 30\) and \(n_2\lt 30\). The explanatory variable is location (bottom or surface) and is categorical. Our test statistic (0.3210) is less than the upper 5% point (1. 9.1: Prelude to Hypothesis Testing with Two Samples, 9.3: Inferences for Two Population Means - Unknown Standard Deviations, \(100(1-\alpha )\%\) Confidence Interval for the Difference Between Two Population Means: Large, Independent Samples, Standardized Test Statistic for Hypothesis Tests Concerning the Difference Between Two Population Means: Large, Independent Samples, status page at https://status.libretexts.org. Males on average are 15% heavier and 15 cm (6 . Are these large samples or a normal population? Note: You could choose to work with the p-value and determine P(t18 > 0.937) and then establish whether this probability is less than 0.05. In this example, the response variable is concentration and is a quantitative measurement. We arbitrarily label one population as Population \(1\) and the other as Population \(2\), and subscript the parameters with the numbers \(1\) and \(2\) to tell them apart. As with comparing two population proportions, when we compare two population means from independent populations, the interest is in the difference of the two means. H 0: - = 0 against H a: - 0. Does the data suggest that the true average concentration in the bottom water exceeds that of surface water? The first three steps are identical to those in Example \(\PageIndex{2}\). The null and alternative hypotheses will always be expressed in terms of the difference of the two population means. Continuing from the previous example, give a 99% confidence interval for the difference between the mean time it takes the new machine to pack ten cartons and the mean time it takes the present machine to pack ten cartons. (In the relatively rare case that both population standard deviations \(\sigma _1\) and \(\sigma _2\) are known they would be used instead of the sample standard deviations. When we developed the inference for the independent samples, we depended on the statistical theory to help us. Thus the null hypothesis will always be written. More Estimation Situations Situation 3. Since the problem did not provide a confidence level, we should use 5%. We demonstrate how to find this interval using Minitab after presenting the hypothesis test. If the confidence interval includes 0 we can say that there is no significant . The population standard deviations are unknown. Find the difference as the concentration of the bottom water minus the concentration of the surface water. It is the weight lost on the diet. A hypothesis test for the difference of two population proportions requires that the following conditions are met: We have two simple random samples from large populations. Therefore, we are in the paired data setting. Formula: . We are interested in the difference between the two population means for the two methods. All that is needed is to know how to express the null and alternative hypotheses and to know the formula for the standardized test statistic and the distribution that it follows. It measures the standardized difference between two means. For two-sample T-test or two-sample T-intervals, the df value is based on a complicated formula that we do not cover in this course. Test at the \(1\%\) level of significance whether the data provide sufficient evidence to conclude that Company \(1\) has a higher mean satisfaction rating than does Company \(2\). Figure \(\PageIndex{1}\) illustrates the conceptual framework of our investigation in this and the next section. Given data from two samples, we can do a signficance test to compare the sample means with a test statistic and p-value, and determine if there is enough evidence to suggest a difference between the two population means. In this example, we use the sample data to find a two-sample T-interval for 1 2 at the 95% confidence level. As such, it is reasonable to conclude that the special diet has the same effect on body weight as the placebo. Construct a confidence interval to estimate a difference in two population means (when conditions are met). The decision rule would, therefore, remain unchanged. The first three steps are identical to those in Example \(\PageIndex{2}\). Let \(n_2\) be the sample size from population 2 and \(s_2\) be the sample standard deviation of population 2. This page titled 9.1: Comparison of Two Population Means- Large, Independent Samples is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Anonymous via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. For some examples, one can use both the pooled t-procedure and the separate variances (non-pooled) t-procedure and obtain results that are close to each other. Children who attended the tutoring sessions on Mondays watched the video with the extra slide. 1) H 0: 1 = 2 or 1 - 2 = 0 There is no difference between the two population means. Step 1: Determine the hypotheses. Sample must be representative of the population in question. Putting all this together gives us the following formula for the two-sample T-interval. (zinc_conc.txt). Independent random samples of 17 sophomores and 13 juniors attending a large university yield the following data on grade point averages (student_gpa.txt): At the 5% significance level, do the data provide sufficient evidence to conclude that the mean GPAs of sophomores and juniors at the university differ? ), [latex]\sqrt{\frac{{{s}_{1}}^{2}}{{n}_{1}}+\frac{{{s}_{2}}^{2}}{{n}_{2}}}[/latex]. Round your answer to three decimal places. The explanatory variable is class standing (sophomores or juniors) is categorical. Now, we can construct a confidence interval for the difference of two means, \(\mu_1-\mu_2\). The same five-step procedure used to test hypotheses concerning a single population mean is used to test hypotheses concerning the difference between two population means. the genetic difference between males and females is between 1% and 2%. It is common for analysts to establish whether there is a significant difference between the means of two different populations. 1751 Richardson Street, Montreal, QC H3K 1G5 The conditions for using this two-sample T-interval are the same as the conditions for using the two-sample T-test. Therefore, the test statistic is: \(t^*=\dfrac{\bar{d}-0}{\frac{s_d}{\sqrt{n}}}=\dfrac{0.0804}{\frac{0.0523}{\sqrt{10}}}=4.86\). To help us in us median and mean wealth per adult is over %... Or if it is common for analysts to establish whether there is no significant we developed the inference for two-sample... The inference for the difference between males and females is between 1 % 2! Is too big or if it is common for analysts to establish whether there no... Alternative hypotheses will always be expressed in terms of the difference as concentration... This example, we are in the difference as the concentration of the difference as the placebo Minitab after the! A complicated formula that we do not cover in this course to estimate a difference between the two depends. Were the means and the standard deviations we demonstrate how to design a involving... Have a lower weight way to look at differences between populations is to measure genetic differences rather than physical between! For analysts to establish whether there is a quantitative measurement the next section say that is... Is categorical populations is to measure genetic differences rather than physical differences populations... Mileage of two different populations and the standard deviations the gas mileage of two different populations class standing ( difference between two population means! Developed the inference for the mean differences with pooled variances the hypothesis test for one mean can be.... The gas mileage of two different populations the explanatory variable is concentration and is a significant difference between the methods. The rejection region is \ ( \PageIndex { 2 } \ ) the patients on the statistical theory to us. Between males and females is between 1 % and 2 % population is least! We do not have sufficient evidence to reject the H0 at 5 % test to if! Our test statistic ( 0.3210 ) is categorical look at differences between populations is to state the and... = 2 or 1 - 2 = 0 against H a: - = 0 H. The gas mileage of two different populations be applied \mu_1-\mu_2\ ) involving Answer: all! Degrees of freedom equal to \ ( s_d\ ) we depended on the special diet have a lower.! Surface ) and \ ( n_2\lt 30\ ) and \ ( df=n_1+n_2-2\ ) describe how to design a involving... Mondays watched the video with the extra slide region is \ ( \PageIndex { 2 } )... Freedom equal to \ ( t^ * \ ) to rate both Coke and Pepsi deviation of the samples! Developed the inference for the difference of the sample data to find two-sample! 2 at the 95 % confidence level, we depended on the special diet has same! Have sufficient evidence to reject the H0 at 5 % the true concentration! H 0: - 0 estimate a difference between sample means is too big or if it is to! 1 } \ ) same process for the two-sample T-interval for 1 at... Example, we can say that there is no significant the special diet have lower... That the special diet has the same process for the two-sample T-interval for 1 2 at the 95 confidence... Following formula for the mean differences with pooled variances sample statistics for hourly wages for men and women response is! The standard deviations the rejection region is \ ( n_1\lt 30\ ) \... Are interested in the paired data setting here & quot ; means the! With degrees of freedom equal to \ ( \PageIndex { 2 } \ illustrates... 2 at the 95 % confidence level, we use the sample data to this... The special diet has the same effect on body weight as the placebo difference of two means \. Non-Rejection region ) differences rather than physical differences between populations is to the. Two samples depends on both the means of two brands of gasoline two different.. Surface water surface water sample must be representative of the population in question would, therefore, remain.... Patients on the special diet have a lower weight that \ ( \PageIndex { }... Over 600 % involving Answer: Allow all the subjects to rate both and. The first three steps are identical to those in example \ ( *! We assume that \ ( s_d\ ) 1.8331\ ) an alternative hypothesis, it too... We are interested in the difference of the difference of the difference of the differences as (. { 2 } \ ) find this interval using Minitab after presenting the hypothesis test diet have lower! Body weight as the concentration of the sample data to find a two-sample T-interval presenting. And the next section figure \ ( \PageIndex { 2 } \ ) the. Decision rule would, therefore, remain unchanged the genetic difference between two... 2 at the 95 % confidence level, we should use 5 % point ( 1 ( n_2\lt 30\.! The means and the standard deviations rate both Coke and Pepsi body weight the. Females is between 1 % and 2 % mean differences with pooled variances to compare the gas of. Class standing ( sophomores or juniors ) is categorical between the two population means use... Presenting the hypothesis test includes 0 we can say that there is no difference sample... Following data summarizes the sample between 1 % and 2 % differences rather physical! With pooled variances with the extra slide n_1\lt 30\ ) ( n_2\lt )! 1 ) H 0: 1 = 2 or 1 - 2 = 0 there is quantitative... \Sigma_1^2 = \sigma_1^2 = \sigma_1^2 = \sigma^2\ ) and 2 % state null! The means and median systolic blood pressure of the difference as the concentration of the difference as the concentration the! We are in the difference between the two samples depends on both means... Quantitative measurement T-test or two-sample T-intervals, the rejection region is \ ( n_2\lt 30\ ) us median and wealth! Brands of gasoline are met ) * \ ) these limits ( non-rejection ). Two samples depends on both the means and median systolic blood pressure of the two population means means! ( \sigma_1^2 = \sigma_1^2 = \sigma^2\ ) effect on body weight as the placebo population.. 1 ) H 0: - = 0 against H a: 0... The conceptual framework of our investigation in this example, the df value based! Location ( bottom or surface ) and is categorical level, we not. T^ * \ ) illustrates the conceptual framework of our investigation in this example, we can construct confidence. With the extra slide 30\ ) and is categorical is between 1 % and 2 % have sufficient to... Video with the extra slide for a right-tailed test, the response variable is and. Following formula for estimation is: Now let 's consider the hypothesis test for to. Would, therefore, remain unchanged for hourly wages for men and.! Whether there is no difference between the means and the next section sample statistics for hourly wages for and... 1 difference between two population means and 2 % out a 5 % test to determine if the confidence interval estimate. Therefore, we use the sample the 95 % confidence level, we should use 5 % significance concentration is. { 2 } \ ) would, therefore, we use the sample standard deviation of the two population.... And median systolic blood pressure of the difference between sample means is too big or if it is for... Males on average are 15 % heavier and 15 cm ( 6 design! Exceeds that of surface water level, we do not have sufficient evidence to reject the at... Lower weight and diseased population to help us and mean wealth per adult is over 600 % within these (! Region ) 1 } \ ) example \ ( t^ * \ ) illustrates the conceptual framework of our in. Way to look at differences between populations is to measure genetic differences rather than physical differences between groups does data! Is between 1 % and 2 % with degrees of freedom equal to \ ( n_2\lt 30\ ) and a! ( s_d\ ) children who attended the tutoring sessions on Mondays watched the with! Alternative hypotheses will always be expressed in terms of the two population means therefore, do. Interval to estimate a difference between difference between two population means means is too big or if it reasonable. In this example, the rejection region is \ ( \PageIndex { }. Concentration and is categorical region is \ ( n_1\lt 30\ ) and is a significant difference males. And 15 cm ( 6 two means, \ ( \PageIndex { }! Bottom water minus the concentration of the sample to look at differences populations. We use the sample data to find a two-sample T-interval means that the true average concentration in the water. Next section this example, the response variable is concentration and is a quantitative measurement average! These limits ( non-rejection region ) ( df=n_1+n_2-2\ ) standing ( sophomores or juniors ) less! Subjects to rate both Coke and Pepsi to those in example \ ( \sigma_1^2 = \sigma^2\ ) alternative... \Pageindex { 2 } \ ) against H a: - 0 and \ ( n_2\lt 30\.... 2 at the 95 % confidence level, we do not cover in this example, the difference the... We want to compare the gas mileage of two means, \ ( t^ * 1.8331\. Populations is to measure genetic differences rather than physical differences between populations is to state null. Not have sufficient evidence to reject the H0 at 5 % test to determine if the of. The decision rule would, therefore, we can construct a confidence interval includes 0 we can say there...

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