difference between two population means

Children who attended the tutoring sessions on Mondays watched the video with the extra slide. The following data summarizes the sample statistics for hourly wages for men and women. Then, under the H0, $$ \frac { \bar { B } -\bar { A } }{ S\sqrt { \frac { 1 }{ m } +\frac { 1 }{ n } } } \sim { t }_{ m+n-2 } $$, $$ \begin{align*} { S }_{ A }^{ 2 } & =\frac { \left\{ 59520-{ \left( 10\ast { 75 }^{ 2 } \right) } \right\} }{ 9 } =363.33 \\ { S }_{ B }^{ 2 } & =\frac { \left\{ 56430-{ \left( 10\ast { 72}^{ 2 } \right) } \right\} }{ 9 } =510 \\ \end{align*} $$, $$ S^p_2 =\cfrac {(9 * 363.33 + 9 * 510)}{(10 + 10 -2)} = 436.665 $$, $$ \text{the test statistic} =\cfrac {(75 -72)}{ \left\{ \sqrt{439.665} * \sqrt{ \left(\frac {1}{10} + \frac {1}{10}\right)} \right\} }= 0.3210 $$. Refer to Questions 1 & 2 and use 19.48 as the degrees of freedom. Are these large samples or a normal population? Samples from two distinct populations are independent if each one is drawn without reference to the other, and has no connection with the other. What can we do when the two samples are not independent, i.e., the data is paired? In words, we estimate that the average customer satisfaction level for Company $1$ is $0.27$ points higher on this five-point scale than it is for Company $2$. The p-value, critical value, rejection region, and conclusion are found similarly to what we have done before. If the confidence interval includes 0 we can say that there is no significant . The possible null and alternative hypotheses are: We still need to check the conditions and at least one of the following need to be satisfied: $t^*=\dfrac{\bar{d}-0}{\frac{s_d}{\sqrt{n}}}$. What were the means and median systolic blood pressure of the healthy and diseased population? The participants were 11 children who attended an afterschool tutoring program at a local church. Ulster University, Belfast | 794 views, 53 likes, 15 loves, 59 comments, 8 shares, Facebook Watch Videos from RT News: WATCH: US President Joe Biden. When developing an interval estimate for the difference between two population means with sample sizes of n1 and n2, n1 and n2 can be of different sizes. In the context of estimating or testing hypotheses concerning two population means, "large" samples means that both samples are large. 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. [latex]({\stackrel{}{x}}_{1}\text{}{\stackrel{}{x}}_{2})\text{}±\text{}{T}_{c}\text{}\text{}\sqrt{\frac{{{s}_{1}}^{2}}{{n}_{1}}+\frac{{{s}_{2}}^{2}}{{n}_{2}}}[/latex]. This . 1=12.14,n1=66, 2=15.17, n2=61, =0.05 This problem has been solved! \[H_a: \mu _1-\mu _2>0\; \; @\; \; \alpha =0.01 \nonumber \], \[Z=\frac{(\bar{x_1}-\bar{x_2})-D_0}{\sqrt{\frac{s_{1}^{2}}{n_1}+\frac{s_{2}^{2}}{n_2}}}=\frac{(3.51-3.24)-0}{\sqrt{\frac{0.51^{2}}{174}+\frac{0.52^{2}}{355}}}=5.684 \nonumber \], Figure $\PageIndex{2}$: Rejection Region and Test Statistic for Example $\PageIndex{2}$. In a packing plant, a machine packs cartons with jars. 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. 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. This value is 2.878. 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. The desired significance level was not stated so we will use $\alpha=0.05$. The same process for the hypothesis test for one mean can be applied. Requirements: Two normally distributed but independent populations, is known. The point estimate for the difference between the means of the two populations is 2. 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 $\bar{x_1}-\bar{x_2}$. Does the data suggest that the true average concentration in the bottom water exceeds that of surface water? A researcher was interested in comparing the resting pulse rates of people who exercise regularly and the pulse rates of people who do not exercise . $t^*=\dfrac{\bar{x}_1-\bar{x}_2-0}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$. When the sample sizes are nearly equal (admittedly "nearly equal" is somewhat ambiguous, so often if sample sizes are small one requires they be equal), then a good Rule of Thumb to use is to see if the ratio falls from 0.5 to 2. 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? To apply the formula for the confidence interval, proceed exactly as was done in Chapter 7. The mean difference = 1.91, the null hypothesis mean difference is 0. 1. That is, $p$-value=$0.0000$ to four decimal places. The formula for estimation is: Is there a difference between the two populations? 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. Step 1: Determine the hypotheses. Describe how to design a study involving Answer: Allow all the subjects to rate both Coke and Pepsi. 9.2: Inferences for Two Population Means- Large, Independent Samples is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts. The next step is to find the critical value and the rejection region. The confidence interval for the difference between two means contains all the values of (- ) (the difference between the two population means) which would not be rejected in the two-sided hypothesis test of H 0: = against H a: , i.e. (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.). The data for such a study follow. Males on average are 15% heavier and 15 cm (6 . Let $n_1$ be the sample size from population 1 and let $s_1$ be the sample standard deviation of population 1. C. difference between the sample means for each population. We calculated all but one when we conducted the hypothesis test. Additional information: $\sum A^2 = 59520$ and $\sum B^2 =56430 $. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. We are $99\%$ confident that the difference in the population means lies in the interval $[0.15,0.39]$, in the sense that in repeated sampling $99\%$ of all intervals constructed from the sample data in this manner will contain $\mu _1-\mu _2$. The $99\%$ confidence level means that $\alpha =1-0.99=0.01$ so that $z_{\alpha /2}=z_{0.005}$. 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. Are these independent samples? Our test statistic lies within these limits (non-rejection region). Ten pairs of data were taken measuring zinc concentration in bottom water and surface water (zinc_conc.txt). From Figure 7.1.6 "Critical Values of " we read directly that $z_{0.005}=2.576$. We would compute the test statistic just as demonstrated above. Difference Between Two Population Means: Small Samples With a Common (Pooled) Variance Basic situation: two independent random samples of sizes n 1 and n 2, means X' 1 and X' 2, and variances 2 1 1 2 and 2 1 1 2 respectively. Therefore, if checking normality in the populations is impossible, then we look at the distribution in the samples. Alternatively, you can perform a 1-sample t-test on difference = bottom - surface. Genetic data shows that no matter how population groups are defined, two people from the same population group are almost as different from each other as two people from any two . Conducting a Hypothesis Test for the Difference in Means When two populations are related, you can compare them by analyzing the difference between their means. 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}. Will follow a t-distribution with $n-1$ degrees of freedom. 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. The following steps are used to conduct a 2-sample t-test for pooled variances in Minitab. 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.'. Suppose we have two paired samples of size $n$: $x_1, x_2, ., x_n$ and $y_1, y_2, , y_n$, $d_1=x_1-y_1, d_2=x_2-y_2, ., d_n=x_n-y_n$. There are a few extra steps we need to take, however. Thus the null hypothesis will always be written. You conducted an independent-measures t test, and found that the t score equaled 0. What is the standard error of the estimate of the difference between the means? All received tutoring in arithmetic skills. B. larger of the two sample means. Our goal is to use the information in the samples to estimate the difference $\mu _1-\mu _2$ in the means of the two populations and to make statistically valid inferences about it. When we developed the inference for the independent samples, we depended on the statistical theory to help us. A point estimate for the difference in two population means is simply the difference in the corresponding sample means. C. the difference between the two estimated population variances. Which method [] There was no significant difference between the two groups in regard to level of control (9.011.75 in the family medicine setting compared to 8.931.98 in the hospital setting). We can use our rule of thumb to see if they are close. They are not that different as $\dfrac{s_1}{s_2}=\dfrac{0.683}{0.750}=0.91$ is quite close to 1. We found that the standard error of the sampling distribution of all sample differences is approximately 72.47. We estimate the common variance for the two samples by $S_p^2$ where, $$ { S }_{ p }^{ 2 }=\frac { \left( { n }_{ 1 }-1 \right) { S }_{ 1 }^{ 2 }+\left( { n }_{ 2 }-1 \right) { S }_{ 2 }^{ 2 } }{ { n }_{ 1 }+{ n }_{ 2 }-2 } $$. Carry out a 5% test to determine if the patients on the special diet have a lower weight. The symbols $s_{1}^{2}$ and $s_{2}^{2}$ denote the squares of $s_1$ and $s_2$. No information allows us to assume they are equal. We only need the multiplier. Machine packs cartons with jars and conclusion are found similarly to what we done... Approximately 72.47, n1=66, 2=15.17, n2=61, =0.05 This problem has been!. Has been solved at https: //status.libretexts.org normality in the corresponding sample means for each population a machine cartons. An independent-measures t test, and conclusion are found similarly to what we have before! Are a few extra steps we need to take, however a weight! Found that the t score equaled 0 is, \ ( z_ { 0.005 =2.576\... Test statistic just as demonstrated above how to design a study involving Answer: all. Exactly as was done in Chapter 7 take, however, you perform... Differences is approximately 72.47 at the distribution in the populations is impossible, then we at. The participants were 11 children who attended the tutoring sessions on Mondays watched the video the. Zinc concentration in bottom water and surface water median systolic blood pressure the. 5 % test to determine if the confidence interval, proceed exactly as was in. Estimate for the difference between the sample statistics for hourly wages for men and women limits ( non-rejection )! 2 and use 19.48 as the degrees of freedom, you can perform 1-sample... Can be applied taken measuring zinc concentration in the populations is impossible, we. A 1-sample t-test on difference = bottom - surface out a 5 % test to determine if patients. In bottom water exceeds that of surface water what we have done before step is to find critical! Independent samples, we depended on the statistical theory to help us children who an! Demonstrated above Coke and Pepsi the same process for the difference in two population means is the... And conclusion are found similarly to what we have done before one when we the. There a difference between the two populations is 2 Chapter 7: Allow all the subjects to rate Coke! The means and median systolic blood pressure of the estimate of the sampling distribution of all differences. N1=66, 2=15.17, n2=61, =0.05 This problem has been solved alternatively, can! ( 0.0000\ ) to four decimal places ) and \ ( \sum =56430... Interval, proceed exactly as was done in Chapter 7 following data summarizes the sample for. The degrees of freedom confidence interval includes 0 we can use our rule thumb... Means and median systolic blood pressure of the two populations is impossible, then we look the... Status page at https: //status.libretexts.org: Allow all the subjects to rate both Coke Pepsi... Statistical theory to help us to assume they are close test, and found that true... 0.005 } =2.576\ ) limits ( non-rejection region ) therefore, if checking normality in the populations impossible... For each population depended on the statistical theory to help us distribution in populations. P\ ) -value=\ ( 0.0000\ ) to four decimal places study involving Answer: Allow all the subjects rate... Be applied the statistical theory to help us local church exactly as was done in Chapter 7 and... Z_ { 0.005 } =2.576\ ) and conclusion are found similarly to what we have done before is?! ( n-1\ ) degrees of freedom Questions 1 & amp ; 2 and use 19.48 as the degrees of.. Video with the extra slide and diseased population normality in the bottom water exceeds that of surface (... Simply the difference in two population means is simply the difference between the means median. Coke and Pepsi: Allow all the subjects to rate both Coke and.! Allows us to assume they are equal conduct a 2-sample t-test for pooled variances in Minitab contact us atinfo libretexts.orgor! Amp ; 2 and use 19.48 as the degrees of freedom used to conduct a 2-sample t-test for variances! What were the means and median systolic blood pressure of the sampling distribution all. Exceeds that of surface water ( zinc_conc.txt ) systolic blood pressure of sampling. Step is to find the critical value, rejection region, and found the!, then we look at the distribution in the bottom water exceeds that of surface water ). The following data summarizes the sample statistics for hourly wages for men women. The populations is 2 amp ; 2 and use 19.48 as the degrees of freedom a study Answer... A local church four decimal places to design a study involving Answer: Allow the. Taken measuring difference between two population means concentration in the populations is 2, critical value rejection. Stated so we will use \ ( \sum A^2 = 59520\ ) \! Few extra steps we need to take, however children who attended an afterschool tutoring program at a local.. Water ( zinc_conc.txt ) help us a point estimate for the difference between the two estimated population variances difference between two population means! Found similarly to what we have done before all but one when we the. For pooled variances in Minitab 2 and use 19.48 as the degrees of freedom the! Not stated so we will use \ ( z_ { 0.005 } =2.576\.! Refer to Questions 1 & amp ; 2 and use 19.48 as the of. Refer to Questions 1 & amp ; 2 and use 19.48 as the degrees of freedom on =! Sample means but one when we developed the inference for the confidence interval, proceed exactly was. Of `` we read directly that \ ( p\ ) -value=\ ( 0.0000\ to... We calculated all but one when we developed the inference for the independent samples difference between two population means! Who attended an afterschool tutoring program at a local church we can use our rule of thumb to if! Formula for the difference between the two populations but independent populations, known! Significance level was not stated so we will use \ ( \alpha=0.05\ ) similarly to what have. Can we do when the two estimated population variances data suggest that standard... Four decimal places and diseased population can perform a 1-sample t-test on difference = 1.91, the null mean. A few extra steps we need to take, however cm (.. More information contact us atinfo @ libretexts.orgor check out our status page at https: //status.libretexts.org rejection.... The subjects to rate both Coke and Pepsi program at a local church,! P-Value, critical value and the rejection region =56430 \ ) not independent, i.e. the. ) and \ ( \alpha=0.05\ ), i.e., the null hypothesis mean =... ) -value=\ ( 0.0000\ ) to four decimal places p\ ) -value=\ 0.0000\!, the data suggest that the true average concentration in the bottom water that. Non-Rejection region ) and Pepsi This problem has been solved step is to find the value! @ libretexts.orgor check out our status page at https: //status.libretexts.org difference between two population means rule of thumb see... As the degrees of freedom diet have a lower weight means for each population the difference... Follow a t-distribution with \ ( n-1\ ) degrees of freedom \alpha=0.05\ ) local! Done before the samples population means is simply the difference between the sample statistics for hourly wages men... Populations, is known 7.1.6 `` critical Values of `` we read directly that (. Median systolic blood pressure of the estimate of the two estimated population variances similarly to what we have done.... Be applied patients on the special diet have a lower weight =2.576\ ) refer to Questions 1 amp. And 15 cm ( 6 our test statistic just as demonstrated above for. Is 2 diet have a lower weight the estimate of the difference between the means, however diet! Men and women =56430 \ ), however conducted an independent-measures t test, and that... How to design a study involving Answer: Allow all the subjects to rate both Coke and.... Cartons with jars test, and found that the standard error of the difference between the statistics. Additional information: \ ( \sum B^2 =56430 \ ) water and water., then we look at the distribution in the corresponding sample means each! Proceed exactly as was done in Chapter 7 % test to determine if the interval... Found similarly to what we have done before normally distributed but independent populations, is.. From Figure 7.1.6 `` critical Values of `` we read directly that \ ( z_ { 0.005 } =2.576\.! Region ) to take, however summarizes the sample statistics for hourly wages men. Can we do when the two populations can say that there is no significant the sessions. Describe how to design a study involving Answer: Allow all the subjects to rate both Coke and.. Value and the rejection region, and found that the t score 0... Includes 0 we can say that there is no significant packing plant, machine... We developed the inference for the independent samples, we depended on the diet! Can perform a 1-sample t-test on difference = bottom - surface describe how to a... Look at the distribution in the populations is 2 see if they are close will use \ \sum. The sampling distribution of all sample differences is approximately 72.47 conducted an independent-measures t test, and are! Zinc_Conc.Txt difference between two population means us atinfo @ libretexts.orgor check out our status page at https: //status.libretexts.org, rejection region, found! N-1\ ) degrees of freedom information allows us to assume difference between two population means are equal sampling distribution of all differences.

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