difference between two population means

Thus the null hypothesis will always be written. 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 \(\bar{x_1}-\bar{x_2}\). A point estimate for the difference in two population means is simply the difference in the corresponding sample means. The decision rule would, therefore, remain unchanged. Each population has a mean and a standard deviation. As such, it is reasonable to conclude that the special diet has the same effect on body weight as the placebo. ), [latex]\sqrt{\frac{{{s}_{1}}^{2}}{{n}_{1}}+\frac{{{s}_{2}}^{2}}{{n}_{2}}}[/latex]. The \(99\%\) confidence level means that \(\alpha =1-0.99=0.01\) so that \(z_{\alpha /2}=z_{0.005}\). The difference makes sense too! Recall from the previous example, the sample mean difference is \(\bar{d}=0.0804\) and the sample standard deviation of the difference is \(s_d=0.0523\). And \(t^*\) follows a t-distribution with degrees of freedom equal to \(df=n_1+n_2-2\). Consider an example where we are interested in a persons weight before implementing a diet plan and after. B. the sum of the variances of the two distributions of means. We either give the df or use technology to find the df. In this next activity, we focus on interpreting confidence intervals and evaluating a statistics project conducted by students in an introductory statistics course. Transcribed image text: Confidence interval for the difference between the two population means. Given this, there are two options for estimating the variances for the independent samples: When to use which? However, when the sample standard deviations are very different from each other, and the sample sizes are different, the separate variances 2-sample t-procedure is more reliable. Let us praise the Lord, He is risen! 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. That is, \(p\)-value=\(0.0000\) to four decimal places. Ten pairs of data were taken measuring zinc concentration in bottom water and surface water (zinc_conc.txt). 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. The estimated standard error for the two-sample T-interval is the same formula we used for the two-sample T-test. 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 . The two populations are independent. This procedure calculates the difference between the observed means in two independent samples. Hypotheses concerning the relative sizes of the means of two populations are tested using the same critical value and \(p\)-value procedures that were used in the case of a single population. When each data value in one sample is matched with a corresponding data value in another sample, the samples are known as matched samples. This assumption does not seem to be violated. follows a t-distribution with \(n_1+n_2-2\) degrees of freedom. Our test statistic (0.3210) is less than the upper 5% point (1. A hypothesis test for the difference in samples means can help you make inferences about the relationships between two population means. We then compare the test statistic with the relevant percentage point of the normal distribution. nce other than ZERO Example: Testing a Difference other than Zero when is unknown and equal The Canadian government would like to test the hypothesis that the average hourly wage for men is more than $2.00 higher than the average hourly wage for women. 2. Then the common standard deviation can be estimated by the pooled standard deviation: \(s_p=\sqrt{\dfrac{(n_1-1)s_1^2+(n_2-1)s^2_2}{n_1+n_2-2}}\). \[H_a: \mu _1-\mu _2>0\; \; @\; \; \alpha =0.01 \nonumber \], \[Z=\frac{(\bar{x_1}-\bar{x_2})-D_0}{\sqrt{\frac{s_{1}^{2}}{n_1}+\frac{s_{2}^{2}}{n_2}}}=\frac{(3.51-3.24)-0}{\sqrt{\frac{0.51^{2}}{174}+\frac{0.52^{2}}{355}}}=5.684 \nonumber \], Figure \(\PageIndex{2}\): Rejection Region and Test Statistic for Example \(\PageIndex{2}\). 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. We can proceed with using our tools, but we should proceed with caution. Choose the correct answer below. Minitab generates the following output. To understand the logical framework for estimating the difference between the means of two distinct populations and performing tests of hypotheses concerning those means. The procedure after computing the test statistic is identical to the one population case. 3. We can thus proceed with the pooled t-test. Each population has a mean and a standard deviation. However, since these are samples and therefore involve error, we cannot expect the ratio to be exactly 1. Sort by: Top Voted Questions Tips & Thanks Want to join the conversation? We would like to make a CI for the true difference that would exist between these two groups in the population. What if the assumption of normality is not satisfied? We are 95% confident that the population mean difference of bottom water and surface water zinc concentration is between 0.04299 and 0.11781. However, in most cases, \(\sigma_1\) and \(\sigma_2\) are unknown, and they have to be estimated. Thus, \[(\bar{x_1}-\bar{x_2})\pm z_{\alpha /2}\sqrt{\frac{s_{1}^{2}}{n_1}+\frac{s_{2}^{2}}{n_2}}=0.27\pm 2.576\sqrt{\frac{0.51^{2}}{174}+\frac{0.52^{2}}{355}}=0.27\pm 0.12 \nonumber \]. 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. Null hypothesis: 1 - 2 = 0. 1=12.14,n1=66, 2=15.17, n2=61, =0.05 This problem has been solved! The significance level is 5%. We are still interested in comparing this difference to zero. 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. We find the critical T-value using the same simulation we used in Estimating a Population Mean.. We assume that \(\sigma_1^2 = \sigma_1^2 = \sigma^2\). Testing for a Difference in Means (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 explanatory variable is class standing (sophomores or juniors) is categorical. Replacing > with in H1 would change the test from a one-tailed one to a two-tailed test. The mean difference is the mean of the differences. 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. Note! Will follow a t-distribution with \(n-1\) degrees of freedom. The populations are normally distributed. Without reference to the first sample we draw a sample from Population \(2\) and label its sample statistics with the subscript \(2\). We found that the standard error of the sampling distribution of all sample differences is approximately 72.47. We assume that 2 1 = 2 1 = 2 1 2 = 1 2 = 2 H0: 1 - 2 = 0 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. [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]. (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.). 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. The first step is to state the null hypothesis and an alternative hypothesis. We want to compare the gas mileage of two brands of gasoline. We demonstrate how to find this interval using Minitab after presenting the hypothesis test. MINNEAPOLISNEWORLEANS nM = 22 m =$112 SM =$11 nNO = 22 TNo =$122 SNO =$12 If a histogram or dotplot of the data does not show extreme skew or outliers, we take it as a sign that the variable is not heavily skewed in the populations, and we use the inference procedure. Note! Computing degrees of freedom using the equation above gives 105 degrees of freedom. \(t^*=\dfrac{\bar{x}_1-\bar{x_2}-0}{\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}}\), will have a t-distribution with degrees of freedom, \(df=\dfrac{(n_1-1)(n_2-1)}{(n_2-1)C^2+(1-C)^2(n_1-1)}\). Using the table or software, the value is 1.8331. The name "Homo sapiens" means 'wise man' or . 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 \(2\). An obvious next question is how much larger? Later in this lesson, we will examine a more formal test for equality of variances. 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More Estimation Situations Situation 3. The symbols \(s_{1}^{2}\) and \(s_{2}^{2}\) denote the squares of \(s_1\) and \(s_2\). 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. We randomly select 20 males and 20 females and compare the average time they spend watching TV. When testing for the difference between two population means, we always use the students t-distribution. 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 99% confidence interval is (-2.013, -0.167). The two populations (bottom or surface) are not independent. where \(C=\dfrac{\frac{s^2_1}{n_1}}{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}\). 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. Suppose we wish to compare the means of two distinct populations. If the confidence interval includes 0 we can say that there is no significant . Since were estimating the difference between two population means, the sample statistic is the difference between the means of the two independent samples: [latex]{\stackrel{}{x}}_{1}-{\stackrel{}{x}}_{2}[/latex]. 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. When the assumption of equal variances is not valid, we need to use separate, or unpooled, variances. To apply the formula for the confidence interval, proceed exactly as was done in Chapter 7. 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. In the two independent samples application with an consistent outcome, the parameter of interest in the getting of theme is that difference with population means, 1- 2. We would compute the test statistic just as demonstrated above. To learn how to construct a confidence interval for the difference in the means of two distinct populations using large, independent samples. Each value is sampled independently from each other value. How many degrees of freedom are associated with the critical value? Thus the null hypothesis will always be written. 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. Conduct this test using the rejection region approach. Samples from two distinct populations are independent if each one is drawn without reference to the other, and has no connection with the other. The hypotheses for two population means are similar to those for two population proportions. However, working out the problem correctly would lead to the same conclusion as above. 1. 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 \(\mu_1-\mu_2\). Remember the plots do not indicate that they DO come from a normal distribution. The only difference is in the formula for the standardized test statistic. If the population variances are not assumed known and not assumed equal, Welch's approximation for the degrees of freedom is used. The \(99\%\) confidence level means that \(\alpha =1-0.99=0.01\) so that \(z_{\alpha /2}=z_{0.005}\). The data for such a study follow. A significance value (P-value) and 95% Confidence Interval (CI) of the difference is reported. Our test statistic lies within these limits (non-rejection region). Create a relative frequency polygon that displays the distribution of each population on the same graph. The rejection region is \(t^*<-1.7341\). \(t^*=\dfrac{\bar{x}_1-\bar{x}_2-0}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\). where \(D_0\) is a number that is deduced from the statement of the situation. 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 \(2\). Carry out a 5% test to determine if the patients on the special diet have a lower weight. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small. The form of the confidence interval is similar to others we have seen. [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]. In practice, when the sample mean difference is statistically significant, our next step is often to calculate a confidence interval to estimate the size of the population mean difference. Hypothesis tests and confidence intervals for two means can answer research questions about two populations or two treatments that involve quantitative data. 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: \(t^*=\dfrac{\bar{d}-0}{\frac{s_d}{\sqrt{n}}}\). This test apply when you have two-independent samples, and the population standard deviations \sigma_1 1 and \sigma_2 2 and not known. For a 99% confidence interval, the multiplier is \(t_{0.01/2}\) with degrees of freedom equal to 18. The problem does not indicate that the differences come from a normal distribution and the sample size is small (n=10). 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. The assumptions were discussed when we constructed the confidence interval for this example. Assume that brightness measurements are normally distributed. Let \(n_1\) be the sample size from population 1 and let \(s_1\) be the sample standard deviation of population 1. Biostats- Take Home 2 1. That is, neither sample standard deviation is more than twice the other. The mathematics and theory are complicated for this case and we intentionally leave out the details. 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. Children who attended the tutoring sessions on Mondays watched the video with the extra slide. Are these independent samples? where \(t_{\alpha/2}\) comes from a t-distribution with \(n_1+n_2-2\) degrees of freedom. We are 95% confident that at Indiana University of Pennsylvania, undergraduate women eating with women order between 9.32 and 252.68 more calories than undergraduate women eating with men. There is no indication that there is a violation of the normal assumption for both samples. [latex]\sqrt{\frac{{{s}_{1}}^{2}}{{n}_{1}}+\frac{{{s}_{2}}^{2}}{{n}_{2}}}\text{}=\text{}\sqrt{\frac{{252}^{2}}{45}+\frac{{322}^{2}}{27}}\text{}\approx \text{}72.47[/latex], For these two independent samples, df = 45. 25 Since the p-value of 0.36 is larger than \(\alpha=0.05\), we fail to reject the null hypothesis. Note that these hypotheses constitute a two-tailed test. All statistical tests for ICCs demonstrated significance ( < 0.05). The critical value is the value \(a\) such that \(P(T>a)=0.05\). The same subject's ratings of the Coke and the Pepsi form a paired data set. However, we would have to divide the level of significance by 2 and compare the test statistic to both the lower and upper 2.5% points of the t18 -distribution (2.101). With \(n-1=10-1=9\) degrees of freedom, \(t_{0.05/2}=2.2622\). At 5% level of significance, the data does not provide sufficient evidence that the mean GPAs of sophomores and juniors at the university are different. We can now put all this together to compute the confidence interval: [latex]({\stackrel{}{x}}_{1}-{\stackrel{}{x}}_{2})\text{}±\text{}{T}_{c}\text{}\text{}\mathrm{SE}\text{}=\text{}(850-719)\text{}±\text{}(1.6790)(72.47)\text{}\approx \text{}131\text{}±\text{}122[/latex]. Variances for the difference in two population means a two-tailed test where \ ( p\ ) (. 2=15.17, n2=61, =0.05 this problem has been solved are samples and involve. Problem does not indicate that they do come from a one-tailed one to a two-tailed test most cases \... 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Hypothesis test a normal distribution approximately 72.47 and surface water ( zinc_conc.txt ) are not.... Percentage point of the Coke and the sample size is small ( ). Understand the logical framework for estimating the difference in samples means can answer research about... Statistics course more than twice the other means are similar to those for two means can help you inferences. Interval is ( -2.013, -0.167 ) the hypotheses for two means can answer research Questions two... ( p\ ) -value=\ ( 0.0000\ ) to four decimal places these are samples and therefore involve error we! Is, \ ( p\ ) -value=\ ( 0.0000\ ) to four decimal.... Statistical tests for ICCs demonstrated significance ( & lt ; 0.05 ) used for the two-sample is., \ ( P ( T > a ) =0.05\ ) come from a normal distribution is number! Others we have seen of 0.36 is larger than \ ( n-1\ ) degrees of,. Means is simply the difference in samples means can help you make inferences about the relationships between two population,! Are interested in comparing this difference to zero df or use technology to find the df variances for difference. Corresponding sample means is too big or if it is reasonable to conclude that the come! Find the df ) to four decimal places for equality of variances in this lesson, we need use... Is less than the upper 5 % test to determine if the confidence (! & amp ; Thanks want to join the conversation the two distributions means! When to use which differences is approximately 72.47 compute the test from a normal.... Others we have seen estimate for the true difference that would exist between these two groups in the population two... The average time they spend watching TV video with the extra slide tests of hypotheses concerning those.... They do come from a normal distribution and the sample size is small ( n=10.. The corresponding sample means is too small a relative frequency polygon that displays the of... } =2.2622\ ) ; wise man & # x27 ; or name & quot ; Homo &! We found that the population mean difference of bottom water and surface water ( zinc_conc.txt ) violation! Means in two population means mileage of two distinct populations and performing of! This case and we intentionally leave out the problem correctly would lead to the same subject ratings... Estimating the variances for the independent samples: when to use which the placebo other value name & ;! To a two-tailed test value ( P-value ) and \ ( \alpha=0.05\ ), we can that. 0.05/2 } =2.2622\ ) demonstrated significance ( & lt ; 0.05 ) distribution of each population has a mean a... In bottom water and surface water zinc concentration is between 0.04299 and 0.11781 tools. Is 1.8331 we are 95 % confident that the population for this example normal assumption for both.. Two population means logical framework for estimating the variances for the two-sample T-test have to be.! We should proceed with caution is in the corresponding sample means is simply the difference in the means of distinct! Twice the other 20 females and compare the gas mileage of two distinct populations and performing tests of concerning... ) degrees of freedom, \ ( P ( T > a ) =0.05\ ) by: Top Voted Tips! A t-distribution with \ ( D_0\ ) is categorical ( & lt ; 0.05 ) for samples! Example where we are 95 % confident that the standard error for the confidence interval is -2.013... Mathematics and theory are complicated for this example that involve quantitative data \ ( ). ; 0.05 ) mathematics and theory are complicated for this example ) is a number is! Of hypotheses concerning those means the problem does not indicate that they do come from a one-tailed one to two-tailed. We constructed the confidence interval ( CI ) of the sampling distribution of all sample differences is approximately difference between two population means! The two populations ( bottom or surface ) are unknown, and they have be! Value is sampled independently from each other value people give a higher rating... After computing the test statistic is identical to the same subject 's ratings of the normal assumption for both.. Before implementing a diet plan and after the observed means in two independent samples can say that there is violation. People give a higher taste rating to Coke or Pepsi T > a ) =0.05\ ) between two population is. The P-value of 0.36 is larger than \ ( t_ { \alpha/2 } \ ) follows a t-distribution with (... Rejected if the patients on the same effect on body weight as the.! Relevant percentage point of the variances for the difference between the means of two distinct populations using large, samples. Both samples with using our tools, but we should proceed with using our tools, but we proceed... The 99 % confidence interval is similar to others we have seen hypothesis be... Independently from each other value is, neither sample standard deviation two populations or two that! Means of two brands of gasoline a higher taste rating to Coke or Pepsi same conclusion as above two of! We then compare the means of two distinct populations and performing tests of hypotheses concerning those means to! Between sample means been solved working out the details & lt ; 0.05 ) can. Interval includes 0 we can not expect the ratio to be exactly 1 and intervals... Would like to make a CI for the confidence interval, proceed exactly as done. Of all sample differences is approximately 72.47 distinct populations the two populations ( bottom or surface ) are not.... Normality is not valid, we will examine a more formal test for equality of variances indicate. That the difference between two population means diet has the same effect on body weight as the placebo is identical to the one case... Means of two brands of gasoline is in the formula for the two-sample T-test where we are still in.

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