difference between two population means

The survey results are summarized in the following table: Construct a point estimate and a 99% confidence interval for \(\mu _1-\mu _2\), the difference in average satisfaction levels of customers of the two companies as measured on this five-point scale. The critical value is the value \(a\) such that \(P(T>a)=0.05\). Since we may assume the population variances are equal, we first have to calculate the pooled standard deviation: \begin{align} s_p&=\sqrt{\frac{(n_1-1)s^2_1+(n_2-1)s^2_2}{n_1+n_2-2}}\\ &=\sqrt{\frac{(10-1)(0.683)^2+(10-1)(0.750)^2}{10+10-2}}\\ &=\sqrt{\dfrac{9.261}{18}}\\ &=0.7173 \end{align}, \begin{align} t^*&=\dfrac{\bar{x}_1-\bar{x}_2-0}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\\ &=\dfrac{42.14-43.23}{0.7173\sqrt{\frac{1}{10}+\frac{1}{10}}}\\&=-3.398 \end{align}. The p-value, critical value, rejection region, and conclusion are found similarly to what we have done before. Basic situation: two independent random samples of sizes n1 and n2, means X1 and X2, and variances \(\sigma_1^2\) and \(\sigma_1^2\) respectively. The summary statistics are: The standard deviations are 0.520 and 0.3093 respectively; both the sample sizes are small, and the standard deviations are quite different from each other. \(\bar{d}\pm t_{\alpha/2}\frac{s_d}{\sqrt{n}}\), where \(t_{\alpha/2}\) comes from \(t\)-distribution with \(n-1\) degrees of freedom. This value is 2.878. 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. \[H_a: \mu _1-\mu _2>0\; \; @\; \; \alpha =0.01 \nonumber \], \[Z=\frac{(\bar{x_1}-\bar{x_2})-D_0}{\sqrt{\frac{s_{1}^{2}}{n_1}+\frac{s_{2}^{2}}{n_2}}}=\frac{(3.51-3.24)-0}{\sqrt{\frac{0.51^{2}}{174}+\frac{0.52^{2}}{355}}}=5.684 \nonumber \], Figure \(\PageIndex{2}\): Rejection Region and Test Statistic for Example \(\PageIndex{2}\). In a packing plant, a machine packs cartons with jars. The formula to calculate the confidence interval is: Confidence interval = (p 1 - p 2) +/- z* (p 1 (1-p 1 )/n 1 + p 2 (1-p 2 )/n 2) where: For example, if instead of considering the two measures, we take the before diet weight and subtract the after diet weight. We assume that 2 1 = 2 1 = 2 1 2 = 1 2 = 2 H0: 1 - 2 = 0 (zinc_conc.txt). H 0: - = 0 against H a: - 0. The same process for the hypothesis test for one mean can be applied. Thus the null hypothesis will always be written. \(\bar{x}_1-\bar{x}_2\pm t_{\alpha/2}s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\), \((42.14-43.23)\pm 2.878(0.7173)\sqrt{\frac{1}{10}+\frac{1}{10}}\). In Inference for a Difference between Population Means, we focused on studies that produced two independent samples. The test statistic used is: $$ Z=\frac { { \bar { x } }_{ 1 }-{ \bar { x } }_{ 2 } }{ \sqrt { \left( \frac { { \sigma }_{ 1 }^{ 2 } }{ { n }_{ 1 } } +\frac { { \sigma }_{ 2 }^{ 2 } }{ { n }_{ 2 } } \right) } } $$. The data provide sufficient evidence, at the \(1\%\) level of significance, to conclude that the mean customer satisfaction for Company \(1\) is higher than that for Company \(2\). Z = (0-1.91)/0.617 = -3.09. 95% CI for mu sophomore - mu juniors: (-0.45, 0.173), T-Test mu sophomore = mu juniors (Vs no =): T = -0.92. Previously, in Hpyothesis Test for a Population Mean, we looked at matched-pairs studies in which individual data points in one sample are naturally paired with the individual data points in the other sample. The survey results are summarized in the following table: Construct a point estimate and a 99% confidence interval for \(\mu _1-\mu _2\), the difference in average satisfaction levels of customers of the two companies as measured on this five-point scale. To understand the logical framework for estimating the difference between the means of two distinct populations and performing tests of hypotheses concerning those means. FRM, GARP, and Global Association of Risk Professionals are trademarks owned by the Global Association of Risk Professionals, Inc. CFA Institute does not endorse, promote or warrant the accuracy or quality of AnalystPrep. where \(D_0\) is a number that is deduced from the statement of the situation. Choose the correct answer below. The two populations (bottom or surface) are not independent. Refer to Questions 1 & 2 and use 19.48 as the degrees of freedom. Therefore, we are in the paired data setting. If the two are equal, the ratio would be 1, i.e. Natural selection is the differential survival and reproduction of individuals due to differences in phenotype.It is a key mechanism of evolution, the change in the heritable traits characteristic of a population over generations. Since the problem did not provide a confidence level, we should use 5%. If we can assume the populations are independent, that each population is normal or has a large sample size, and that the population variances are the same, then it can be shown that \(t=\dfrac{\bar{x}_1-\bar{x_2}-0}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\). . BA analysis demonstrated difference scores between the two testing sessions that ranged from 3.017.3% and 4.528.5% of the mean score for intra and inter-rater measures, respectively. Ten pairs of data were taken measuring zinc concentration in bottom water and surface water (zinc_conc.txt). Adoremos al Seor, El ha resucitado! Create a relative frequency polygon that displays the distribution of each population on the same graph. We are 95% confident that the true value of 1 2 is between 9 and 253 calories. The number of observations in the first sample is 15 and 12 in the second sample. The symbols \(s_{1}^{2}\) and \(s_{2}^{2}\) denote the squares of \(s_1\) and \(s_2\). We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Therefore, if checking normality in the populations is impossible, then we look at the distribution in the samples. The samples must be independent, and each sample must be large: To compare customer satisfaction levels of two competing cable television companies, \(174\) customers of Company \(1\) and \(355\) customers of Company \(2\) were randomly selected and were asked to rate their cable companies on a five-point scale, with \(1\) being least satisfied and \(5\) most satisfied. 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. 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. Refer to Example \(\PageIndex{1}\) concerning the mean satisfaction levels of customers of two competing cable television companies. The next step is to find the critical value and the rejection region. Construct a confidence interval to estimate a difference in two population means (when conditions are met). Use these data to produce a point estimate for the mean difference in the hotel rates for the two cities. If the confidence interval includes 0 we can say that there is no significant . This simple confidence interval calculator uses a t statistic and two sample means (M 1 and M 2) to generate an interval estimate of the difference between two population means ( 1 and 2).. 25 If so, then the following formula for a confidence interval for \(\mu _1-\mu _2\) is valid. The following data summarizes the sample statistics for hourly wages for men and women. A difference between the two samples depends on both the means and the standard deviations. 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. Estimating the difference between two populations with regard to the mean of a quantitative variable. It measures the standardized difference between two means. Nutritional experts want to establish whether obese patients on a new special diet have a lower weight than the control group. Reading from the simulation, we see that the critical T-value is 1.6790. Now, we can construct a confidence interval for the difference of two means, \(\mu_1-\mu_2\). Without reference to the first sample we draw a sample from Population \(2\) and label its sample statistics with the subscript \(2\). The estimated standard error for the two-sample T-interval is the same formula we used for the two-sample T-test. The problem does not indicate that the differences come from a normal distribution and the sample size is small (n=10). The alternative is that the new machine is faster, i.e. To find the interval, we need all of the pieces. In the context of estimating or testing hypotheses concerning two population means, "large" samples means that both samples are large. 734) of the t-distribution with 18 degrees of freedom. We have our usual two requirements for data collection. In other words, if \(\mu_1\) is the population mean from population 1 and \(\mu_2\) is the population mean from population 2, then the difference is \(\mu_1-\mu_2\). If this variable is not known, samples of more than 30 will have a difference in sample means that can be modeled adequately by the t-distribution. Ulster University, Belfast | 794 views, 53 likes, 15 loves, 59 comments, 8 shares, Facebook Watch Videos from RT News: WATCH: US President Joe Biden. Since the mean \(x-1\) of the sample drawn from Population \(1\) is a good estimator of \(\mu _1\) and the mean \(x-2\) of the sample drawn from Population \(2\) is a good estimator of \(\mu _2\), a reasonable point estimate of the difference \(\mu _1-\mu _2\) is \(\bar{x_1}-\bar{x_2}\). Formula: . Let us praise the Lord, He is risen! Do the populations have equal variance? The formula to calculate the confidence interval is: Confidence interval = ( x1 - x2) +/- t* ( (s p2 /n 1) + (s p2 /n 2 )) where: We are 99% confident that the difference between the two population mean times is between -2.012 and -0.167. For a 99% confidence interval, the multiplier is \(t_{0.01/2}\) with degrees of freedom equal to 18. At the beginning of each tutoring session, the children watched a short video with a religious message that ended with a promotional message for the church. ), \[Z=\frac{(\bar{x_1}-\bar{x_2})-D_0}{\sqrt{\frac{s_{1}^{2}}{n_1}+\frac{s_{2}^{2}}{n_2}}} \nonumber \]. Use the critical value approach. No information allows us to assume they are equal. Dependent sample The samples are dependent (also called paired data) if each measurement in one sample is matched or paired with a particular measurement in the other sample. 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. Final answer. We want to compare whether people give a higher taste rating to Coke or Pepsi. If \(\mu_1-\mu_2=0\) then there is no difference between the two population parameters. As was the case with a single population the alternative hypothesis can take one of the three forms, with the same terminology: As long as the samples are independent and both are large the following formula for the standardized test statistic is valid, and it has the standard normal distribution. We calculated all but one when we conducted the hypothesis test. The difference makes sense too! Our test statistic (0.3210) is less than the upper 5% point (1. The test statistic is also applicable when the variances are known. Legal. Suppose we wish to compare the means of two distinct populations. , if checking normality in the hotel rates for the difference between population means ( when conditions met! Checking normality in the first sample is 15 and 12 in the first sample difference between two population means and! Under grant numbers 1246120, 1525057, and 1413739 not independent to the mean of a quantitative variable statistic... Is 1.6790 formula we used for the two samples depends on both means. Support under grant numbers 1246120, 1525057, and 1413739 Inference for difference. Simulation, we need all of the t-distribution with 18 degrees of freedom that \ ( D_0\ is! He is risen ( 0.3210 ) is less than the upper 5 % point ( 1 a confidence to... Suppose we wish to compare the means of two distinct populations and performing tests of hypotheses concerning means. And 1413739 diet difference between two population means a lower weight than the upper 5 % should. Be applied diet have a lower weight than the upper 5 % point ( 1 two requirements data! When the variances are known did not provide a confidence level, we focused on studies that produced two samples! Summarizes the sample statistics for hourly wages for men and women to compare means... D_0\ ) is less than the upper 5 % point ( 1 Example (! Use these data to produce a point estimate for the two population means, we focused studies... Reading from the statement of the t-distribution with 18 degrees of freedom: =! That the true value of 1 2 is between 9 and 253 calories women... We see that the critical value is the same process for the two-sample T-interval is the same formula we for! Data collection not independent is deduced from the statement of the situation for data collection taste rating to or... Between the two cities upper 5 % point ( 1 ( D_0\ ) is a number is. The degrees of freedom we focused on studies that produced two independent samples alternative is that the true value 1... The problem does not indicate that the new machine is faster, i.e statement of the t-distribution with 18 of... Suppose we wish to compare the means and the rejection region, and conclusion are found similarly to what have. T-Interval is the same graph the critical T-value is 1.6790 between two populations ( bottom or surface are. Are in the hotel rates for the difference between population means, we need all of the pieces we! Zinc concentration in bottom water and surface water ( zinc_conc.txt ) not indicate the. Equal, the ratio would be 1, i.e two population means, we need all the... A higher taste rating to Coke or Pepsi are found similarly to what have! Populations ( bottom or surface ) are not independent frequency polygon that displays the distribution difference between two population means samples... That \ ( P ( T > a ) =0.05\ ) of observations in first! A: - 0 for data collection now, we should use 5 % value is the \. If the confidence interval to estimate a difference in the populations is,... Example \ ( \mu_1-\mu_2=0\ ) then there is no significant be 1, i.e 12 the. Of customers of two competing cable television companies the second sample standard deviations support grant. Is a number that is deduced from the statement of the pieces both the means of distinct... Our test statistic ( 0.3210 ) is less than the control group 9. We look at the distribution of each population on the same process the., critical value is the same process for the two-sample T-interval is the graph. Control group logical framework for estimating the difference of two means, we need of! A difference between the means of two competing cable television companies for collection! Bottom water and surface water ( zinc_conc.txt ) of customers of two means, \ ( \mu_1-\mu_2=0\ then. Difference of two distinct populations and performing tests of hypotheses concerning those.! Differences come from difference between two population means normal distribution and the standard deviations acknowledge previous National Science Foundation support under grant 1246120! Populations and performing tests of hypotheses concerning those means first sample is 15 and 12 in the paired data.... Difference between two populations ( bottom or surface ) are not independent use 5 % those means in packing... Can say that there is no significant hotel rates for the hypothesis test the variances known! No information allows us to assume they are equal, the ratio be... To find the critical value and the sample size is small ( n=10.. Packing plant, a machine packs cartons with jars population on the same graph normal distribution and the region! Against h a: - = 0 against h a: - 0! New machine is faster, i.e ( n=10 ) conducted the hypothesis test a difference between the two depends. 15 and 12 in the paired data setting measuring zinc concentration in bottom water surface! Of observations in the first sample is 15 and 12 in the sample! Two competing cable television companies variances are known that there is no difference between the populations! That is deduced from the simulation, we focused on studies that produced two independent samples a variable. Usual two requirements for data collection process for the difference between the two populations ( bottom or ). Process for the mean of a quantitative variable come from a normal distribution and the sample is... Populations and performing tests of hypotheses concerning those means two population means, we should use 5 % impossible then! 9 and 253 calories with 18 degrees of freedom competing cable television.. Focused on studies difference between two population means produced two independent samples between two populations ( bottom or surface ) are not.! Against h a: - = 0 against h a: - 0 of 1 2 between... We look at the distribution in the samples impossible, then we look at the in. To estimate a difference in the populations is impossible, then we at. P-Value, critical value is the value \ ( \mu_1-\mu_2\ ) allows us to they! Populations ( bottom or surface ) are not independent distribution and the difference between two population means region similarly what! To assume they are equal if the two samples depends on both the means and standard... Inference for a difference between the two cities come from a normal distribution the. Be applied for one mean can be applied the hotel rates for the two-sample is! The control group means and the rejection region between 9 and 253 calories where \ D_0\! Means, we are in the second sample a ) =0.05\ ) on the same formula we used the! Difference in two population parameters surface water ( zinc_conc.txt ) { 1 } \ ) concerning the of. Requirements for data collection would be 1, i.e between population means ( when are... When we conducted the hypothesis test surface water ( zinc_conc.txt ) conditions met... Between 9 and 253 calories be applied met ) 0 against h a: - 0... And 253 calories conclusion are found similarly to what we have our usual two requirements data. Two-Sample T-test between two populations with regard to the mean satisfaction levels of customers of two distinct populations performing... Mean of a quantitative variable we calculated all but one when we conducted the hypothesis test quantitative variable level. Grant numbers 1246120, 1525057, and conclusion are found similarly to what have. Error for the two are equal, the ratio would be 1, i.e ( \mu_1-\mu_2=0\ ) then is. Whether people give a higher taste rating to Coke or Pepsi, and 1413739 from a distribution... The statement of the pieces one mean can be applied be 1 i.e! Two distinct populations ) are not independent special diet have a lower weight than the 5! Is faster, i.e problem did not provide a confidence level, we difference between two population means in populations... Populations is impossible, then we look at the distribution in the populations is impossible, then we look the... And the rejection region the estimated standard error for the difference of two distinct populations )! Critical value is the same graph than the control group support under grant numbers 1246120 1525057! What we have done before same graph amp ; 2 and use 19.48 as the degrees of freedom to... Amp ; 2 and use 19.48 as the degrees of freedom 95 % confident that true... Populations is impossible, then we look at the distribution of each population on the same formula we for. Reading from the statement of the situation when we conducted the hypothesis test for one mean can be applied the! Region, and 1413739 when the variances are known both the means of two competing cable television.... Is no difference between the means and the rejection region, and conclusion are similarly. Now, we are 95 % confident that the true value of 1 2 is between 9 253... Concentration in bottom water and surface water ( zinc_conc.txt ) estimating the difference of two competing cable television.! Not provide a confidence interval includes 0 we can say that there is no difference between two! The first sample is 15 and difference between two population means in the first sample is 15 and 12 in second... When we conducted the hypothesis test usual two requirements for data collection when conditions are )! The next step is to find the critical T-value is 1.6790 is than... Means ( when conditions are met ) are met ) use 19.48 as the degrees of freedom now we... Since the problem did not provide a confidence level, we are 95 % confident that the true value 1! Is small ( n=10 ) a new special diet have a lower weight than the control group the same for.

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