difference between two population means
The following steps are used to conduct a 2-sample t-test for pooled variances in Minitab. What can we do when the two samples are not independent, i.e., the data is paired? However, in most cases, \(\sigma_1\) and \(\sigma_2\) are unknown, and they have to be estimated. D Suppose that populations of men and women have the following summary statistics for their heights (in centimeters): Mean Standard deviation Men = 172 M =172mu, start subscript, M, end subscript, equals, 172 = 7.2 M =7.2sigma, start subscript, M, end subscript, equals, 7, point, 2 Women = 162 W =162mu, start subscript, W, end subscript, equals, 162 = 5.4 W =5.4sigma, start . 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. The children ranged in age from 8 to 11. If \(\bar{d}\) is normal (or the sample size is large), the sampling distribution of \(\bar{d}\) is (approximately) normal with mean \(\mu_d\), standard error \(\dfrac{\sigma_d}{\sqrt{n}}\), and estimated standard error \(\dfrac{s_d}{\sqrt{n}}\). The results, (machine.txt), in seconds, are shown in the tables. 9.2: Comparison of Two Population Means - Small, Independent Samples, \(100(1-\alpha )\%\) 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, source@https://2012books.lardbucket.org/books/beginning-statistics, status page at https://status.libretexts.org. After 6 weeks, the average weight of 10 patients (group A) on the special diet is 75kg, while that of 10 more patients of the control group (B) is 72kg. The hypotheses for two population means are similar to those for two population proportions. In this example, we use the sample data to find a two-sample T-interval for 1 2 at the 95% confidence level. Children who attended the tutoring sessions on Mondays watched the video with the extra slide. When the assumption of equal variances is not valid, we need to use separate, or unpooled, variances. Interpret the confidence interval in context. All received tutoring in arithmetic skills. where \(t_{\alpha/2}\) comes from a t-distribution with \(n_1+n_2-2\) degrees of freedom. Sample must be representative of the population in question. 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. Let's take a look at the normality plots for this data: From the normal probability plots, we conclude that both populations may come from normal distributions. support@analystprep.com. The difference between the two sample proportions is 0.63 - 0.42 = 0.21. In Minitab, if you choose a lower-tailed or an upper-tailed hypothesis test, an upper or lower confidence bound will be constructed, respectively, rather than a confidence interval. The same subject's ratings of the Coke and the Pepsi form a paired data set. Thus the null hypothesis will always be written. Now we can apply all we learned for the one sample mean to the difference (Cool!). The following dialog boxes will then be displayed. Therefore, we are in the paired data setting. Hypothesis test. where \(D_0\) is a number that is deduced from the statement of the situation. We found that the standard error of the sampling distribution of all sample differences is approximately 72.47. 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. Therefore, if checking normality in the populations is impossible, then we look at the distribution in the samples. The following options can be given: Remember, the default for the 2-sample t-test in Minitab is the non-pooled one. Since 0 is not in our confidence interval, then the means are statistically different (or statistical significant or statistically different). There are a few extra steps we need to take, however. 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. The null and alternative hypotheses will always be expressed in terms of the difference of the two population means. We are 99% confident that the difference between the two population mean times is between -2.012 and -0.167. Yes, since the samples from the two machines are not related. Since the population standard deviations are unknown, we can use the t-distribution and the formula for the confidence interval of the difference between two means with independent samples: (ci lower, ci upper) = (x - x) t (/2, df) * s_p * sqrt (1/n + 1/n) where x and x are the sample means, s_p is the pooled . 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 \(2\). 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. The conditions for using this two-sample T-interval are the same as the conditions for using the two-sample T-test. 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}. We are 95% confident that the difference between the mean GPA of sophomores and juniors is between -0.45 and 0.173. (Assume that the two samples are independent simple random samples selected from normally distributed populations.) 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. Alternatively, you can perform a 1-sample t-test on difference = bottom - surface. The name "Homo sapiens" means 'wise man' or . Suppose we replace > with in H1 in the example above, would the decision rule change? We assume that 2 1 = 2 1 = 2 1 2 = 1 2 = 2 H0: 1 - 2 = 0 \(t^*=\dfrac{\bar{x}_1-\bar{x_2}-0}{\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}}\), will have a t-distribution with degrees of freedom, \(df=\dfrac{(n_1-1)(n_2-1)}{(n_2-1)C^2+(1-C)^2(n_1-1)}\). For instance, they might want to know whether the average returns for two subsidiaries of a given company exhibit a significant difference. Perform the 2-sample t-test in Minitab with the appropriate alternative hypothesis. The symbols \(s_{1}^{2}\) and \(s_{2}^{2}\) denote the squares of \(s_1\) and \(s_2\). 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. Let \(\mu_1\) denote the mean for the new machine and \(\mu_2\) denote the mean for the old machine. Samples from two distinct populations are independent if each one is drawn without reference to the other, and has no connection with the other. When each data value in one sample is matched with a corresponding data value in another sample, the samples are known as matched samples. Perform the test of Example \(\PageIndex{2}\) using the \(p\)-value approach. Describe how to design a study involving Answer: Allow all the subjects to rate both Coke and Pepsi. The students were inspired by a similar study at City University of New York, as described in David Moores textbook The Basic Practice of Statistics (4th ed., W. H. Freeman, 2007). The result is a confidence interval for the difference between two population means, In the context of estimating or testing hypotheses concerning two population means, "large" samples means that both samples are large. As above, the null hypothesis tends to be that there is no difference between the means of the two populations; or, more formally, that the difference is zero (so, for example, that there is no difference between the average heights of two populations of . 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. Adoremos al Seor, El ha resucitado! 3. This assumption is called the assumption of homogeneity of variance. In this next activity, we focus on interpreting confidence intervals and evaluating a statistics project conducted by students in an introductory statistics course. Final answer. The \(99\%\) confidence level means that \(\alpha =1-0.99=0.01\) so that \(z_{\alpha /2}=z_{0.005}\). Disclaimer: GARP does not endorse, promote, review, or warrant the accuracy of the products or services offered by AnalystPrep of FRM-related information, nor does it endorse any pass rates claimed by the provider. As such, it is reasonable to conclude that the special diet has the same effect on body weight as the placebo. Does the data suggest that the true average concentration in the bottom water exceeds that of surface water? With \(n-1=10-1=9\) degrees of freedom, \(t_{0.05/2}=2.2622\). The samples must be independent, and each sample must be large: \(n_1\geq 30\) and \(n_2\geq 30\). Therefore, the test statistic is: \(t^*=\dfrac{\bar{d}-0}{\frac{s_d}{\sqrt{n}}}=\dfrac{0.0804}{\frac{0.0523}{\sqrt{10}}}=4.86\). 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 . What conditions are necessary in order to use a t-test to test the differences between two population means? The mean difference = 1.91, the null hypothesis mean difference is 0. When the sample sizes are small, the estimates may not be that accurate and one may get a better estimate for the common standard deviation by pooling the data from both populations if the standard deviations for the two populations are not that different. The alternative hypothesis, Ha, takes one of the following three forms: As usual, how we collect the data determines whether we can use it in the inference procedure. Our test statistic lies within these limits (non-rejection region). 113K views, 2.8K likes, 58 loves, 140 comments, 1.2K shares, Facebook Watch Videos from : # # #____ ' . We randomly select 20 couples and compare the time the husbands and wives spend watching TV. The following data summarizes the sample statistics for hourly wages for men and women. where \(D_0\) is a number that is deduced from the statement of the situation. 105 Question 32: For a test of the equality of the mean returns of two non-independent populations based on a sample, the numerator of the appropriate test statistic is the: A. average difference between pairs of returns. It only shows if there are clear violations. The mathematics and theory are complicated for this case and we intentionally leave out the details. We are still interested in comparing this difference to zero. Welch, B. L. (1938). 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 \(t=\dfrac{\bar{x}_1-\bar{x_2}-0}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\). 1=12.14,n1=66, 2=15.17, n2=61, =0.05 This problem has been solved! Another way to look at differences between populations is to measure genetic differences rather than physical differences between groups. 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.'. In this section, we are going to approach constructing the confidence interval and developing the hypothesis test similarly to how we approached those of the difference in two proportions. man, woman | 1.2K views, 15 likes, 0 loves, 1 comments, 2 shares, Facebook Watch Videos from DrPhil Show 2023: Dr Phil Show 2023 The Cougar Controversy Older Woman Dating Younger Men First, we need to consider whether the two populations are independent. 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. Describe how to design a study involving independent sample and dependent samples. Refer to Example \(\PageIndex{1}\) concerning the mean satisfaction levels of customers of two competing cable television companies. Also assume that the population variances are unequal. Relationship between population and sample: A population is the entire group of individuals or objects that we want to study, while a sample is a subset of the population that is used to make inferences about the population. More Estimation Situations Situation 3. That is, \(p\)-value=\(0.0000\) to four decimal places. Is this an independent sample or paired sample? The formula for estimation is: We then compare the test statistic with the relevant percentage point of the normal distribution. The following are examples to illustrate the two types of samples. If the difference was defined as surface - bottom, then the alternative would be left-tailed. The first three steps are identical to those in Example \(\PageIndex{2}\). The two populations (bottom or surface) are not independent. Trace metals in drinking water affect the flavor and an unusually high concentration can pose a health hazard. 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. Ten pairs of data were taken measuring zinc concentration in bottom water and surface water (zinc_conc.txt). If \(\mu_1-\mu_2=0\) then there is no difference between the two population parameters. When we developed the inference for the independent samples, we depended on the statistical theory to help us. The two types of samples require a different theory to construct a confidence interval and develop a hypothesis test. The drinks should be given in random order. 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. 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. That is, neither sample standard deviation is more than twice the other. The null and alternative hypotheses will always be expressed in terms of the difference of the two population means. Suppose we wish to compare the means of two distinct populations. 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: \(t^*=\dfrac{\bar{d}-0}{\frac{s_d}{\sqrt{n}}}\). From Figure 7.1.6 "Critical Values of " we read directly that \(z_{0.005}=2.576\). Formula: . There is no indication that there is a violation of the normal assumption for both samples. Computing degrees of freedom using the equation above gives 105 degrees of freedom. We are 95% confident that the true value of 1 2 is between 9 and 253 calories. Wed love your input. 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 \(2\). A point estimate for the difference in two population means is simply the difference in the corresponding sample means. The significance level is 5%. Independent Samples Confidence Interval Calculator. The Minitab output for the packing time example: Equal variances are assumed for this analysis. The difference between the two values is due to the fact that our population includes military personnel from D.C. which accounts for 8,579 of the total number of military personnel reported by the US Census Bureau.\n\nThe value of the standard deviation that we calculated in Exercise 8a is 16. Let us praise the Lord, He is risen! Assume that the population variances are equal. \(\bar{d}\pm t_{\alpha/2}\frac{s_d}{\sqrt{n}}\), where \(t_{\alpha/2}\) comes from \(t\)-distribution with \(n-1\) degrees of freedom. Which method [] Hypothesis tests and confidence intervals for two means can answer research questions about two populations or two treatments that involve quantitative data. Thus, \[(\bar{x_1}-\bar{x_2})\pm z_{\alpha /2}\sqrt{\frac{s_{1}^{2}}{n_1}+\frac{s_{2}^{2}}{n_2}}=0.27\pm 2.576\sqrt{\frac{0.51^{2}}{174}+\frac{0.52^{2}}{355}}=0.27\pm 0.12 \nonumber \]. We estimate the common variance for the two samples by \(S_p^2\) where, $$ { S }_{ p }^{ 2 }=\frac { \left( { n }_{ 1 }-1 \right) { S }_{ 1 }^{ 2 }+\left( { n }_{ 2 }-1 \right) { S }_{ 2 }^{ 2 } }{ { n }_{ 1 }+{ n }_{ 2 }-2 } $$. Ulster University, Belfast | 794 views, 53 likes, 15 loves, 59 comments, 8 shares, Facebook Watch Videos from RT News: WATCH: US President Joe Biden. Estimating the Difference in Two Population Means Learning outcomes Construct a confidence interval to estimate a difference in two population means (when conditions are met). Then, under the H0, $$ \frac { \bar { B } -\bar { A } }{ S\sqrt { \frac { 1 }{ m } +\frac { 1 }{ n } } } \sim { t }_{ m+n-2 } $$, $$ \begin{align*} { S }_{ A }^{ 2 } & =\frac { \left\{ 59520-{ \left( 10\ast { 75 }^{ 2 } \right) } \right\} }{ 9 } =363.33 \\ { S }_{ B }^{ 2 } & =\frac { \left\{ 56430-{ \left( 10\ast { 72}^{ 2 } \right) } \right\} }{ 9 } =510 \\ \end{align*} $$, $$ S^p_2 =\cfrac {(9 * 363.33 + 9 * 510)}{(10 + 10 -2)} = 436.665 $$, $$ \text{the test statistic} =\cfrac {(75 -72)}{ \left\{ \sqrt{439.665} * \sqrt{ \left(\frac {1}{10} + \frac {1}{10}\right)} \right\} }= 0.3210 $$. The first three steps are identical to those in Example \(\PageIndex{2}\). 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. H 0: - = 0 against H a: - 0. 1751 Richardson Street, Montreal, QC H3K 1G5 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. Z = (0-1.91)/0.617 = -3.09. The confidence interval gives us a range of reasonable values for the difference in population means 1 2. O A. The participants were 11 children who attended an afterschool tutoring program at a local church. The response variable is GPA and is quantitative. 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 . ), [latex]\sqrt{\frac{{{s}_{1}}^{2}}{{n}_{1}}+\frac{{{s}_{2}}^{2}}{{n}_{2}}}[/latex]. (zinc_conc.txt). The decision rule would, therefore, remain unchanged. We use the two-sample hypothesis test and confidence interval when the following conditions are met: [latex]({\stackrel{}{x}}_{1}\text{}\text{}\text{}{\stackrel{}{x}}_{2})\text{}±\text{}{T}_{c}\text{}\text{}\sqrt{\frac{{{s}_{1}}^{2}}{{n}_{1}}+\frac{{{s}_{2}}^{2}}{{n}_{2}}}[/latex], [latex]T\text{}=\text{}\frac{(\mathrm{Observed}\text{}\mathrm{difference}\text{}\mathrm{in}\text{}\mathrm{sample}\text{}\mathrm{means})\text{}-\text{}(\mathrm{Hypothesized}\text{}\mathrm{difference}\text{}\mathrm{in}\text{}\mathrm{population}\text{}\mathrm{means})}{\mathrm{Standard}\text{}\mathrm{error}}[/latex], [latex]T\text{}=\text{}\frac{({\stackrel{}{x}}_{1}-{\stackrel{}{x}}_{2})\text{}-\text{}({}_{1}-{}_{2})}{\sqrt{\frac{{{s}_{1}}^{2}}{{n}_{1}}+\frac{{{s}_{2}}^{2}}{{n}_{2}}}}[/latex], We use technology to find the degrees of freedom to determine P-values and critical t-values for confidence intervals. (The actual value is approximately \(0.000000007\).). 734) of the t-distribution with 18 degrees of freedom. 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. Biostats- Take Home 2 1. A point estimate for the difference in two population means is simply the difference in the corresponding sample means. 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 \(\bar{x_1}-\bar{x_2}\). To use the methods we developed previously, we need to check the conditions. The null theory is always that there is no difference between groups with respect to means, i.e., The null thesis can also becoming written as being: H 0: 1 = 2. Nutritional experts want to establish whether obese patients on a new special diet have a lower weight than the control group. For example, if instead of considering the two measures, we take the before diet weight and subtract the after diet weight. A confidence interval for the difference in two population means is computed using a formula in the same fashion as was done for a single population mean. \(H_0\colon \mu_1-\mu_2=0\) vs \(H_a\colon \mu_1-\mu_2\ne0\). In a packing plant, a machine packs cartons with jars. 95% CI for mu sophomore - mu juniors: (-0.45, 0.173), T-Test mu sophomore = mu juniors (Vs no =): T = -0.92. The population standard deviations are unknown. Interpret the confidence interval in context. It is supposed that a new machine will pack faster on the average than the machine currently used. Use these data to produce a point estimate for the mean difference in the hotel rates for the two cities. In ecology, the occupancy-abundance (O-A) relationship is the relationship between the abundance of species and the size of their ranges within a region. To learn how to perform a test of hypotheses concerning the difference between the means of two distinct populations using large, independent samples. Alternative hypothesis: 1 - 2 0. The test statistic has the standard normal distribution. The assumptions were discussed when we constructed the confidence interval for this example. 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 \(2\). This is made possible by the central limit theorem. B. the sum of the variances of the two distributions of means. Samples must be random in order to remove or minimize bias. Confidence Interval to Estimate 1 2 the genetic difference between males and females is between 1% and 2%. In Inference for a Difference between Population Means, we focused on studies that produced two independent samples. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small. We have \(n_1\lt 30\) and \(n_2\lt 30\). How many degrees of freedom are associated with the critical value? Step 1: Determine the hypotheses. (The actual value is approximately \(0.000000007\).). Standard deviation is 0.617. Refer to Question 1. We would like to make a CI for the true difference that would exist between these two groups in the population. 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: A hypothesis test for the difference in samples means can help you make inferences about the relationships between two population means. Then the common standard deviation can be estimated by the pooled standard deviation: \(s_p=\sqrt{\dfrac{(n_1-1)s_1^2+(n_2-1)s^2_2}{n_1+n_2-2}}\). We want to compare whether people give a higher taste rating to Coke or Pepsi. The experiment lasted 4 weeks. The point estimate of \(\mu _1-\mu _2\) is, \[\bar{x_1}-\bar{x_2}=3.51-3.24=0.27 \nonumber \]. Introductory Statistics (Shafer and Zhang), { "9.01:_Comparison_of_Two_Population_Means-_Large_Independent_Samples" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.
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