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. We randomly select 20 couples and compare the time the husbands and wives spend watching TV. This value is 2.878. The theory, however, required the samples to be independent. To apply the formula for the confidence interval, proceed exactly as was done in Chapter 7. Note! 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. The alternative is that the new machine is faster, i.e. 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. 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 value of our test statistic falls in the rejection region. 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. Using the p-value to draw a conclusion about our example: Reject\(H_0\) and conclude that bottom zinc concentration is higher than surface zinc concentration. 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 . The test statistic has the standard normal distribution. If so, then the following formula for a confidence interval for \(\mu _1-\mu _2\) is valid. Each value is sampled independently from each other value. 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. 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. At this point, the confidence interval will be the same as that of one sample. \(\frac{s_1}{s_2}=1\). The critical value is -1.7341. The problem does not indicate that the differences come from a normal distribution and the sample size is small (n=10). \(\bar{d}\pm t_{\alpha/2}\frac{s_d}{\sqrt{n}}\), where \(t_{\alpha/2}\) comes from \(t\)-distribution with \(n-1\) degrees of freedom. 9.2: Comparison off Two Population Means . The symbols \(s_{1}^{2}\) and \(s_{2}^{2}\) denote the squares of \(s_1\) and \(s_2\). 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. 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 alternative is left-tailed so the critical value is the value \(a\) such that \(P(T 0): T-Value = 4.86 P-Value = 0.000. the genetic difference between males and females is between 1% and 2%. 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. H 0: - = 0 against H a: - 0. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Therefore, the test statistic is: \(t^*=\dfrac{\bar{d}-0}{\frac{s_d}{\sqrt{n}}}=\dfrac{0.0804}{\frac{0.0523}{\sqrt{10}}}=4.86\). As we learned in the previous section, if we consider the difference rather than the two samples, then we are back in the one-sample mean scenario. 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. To avoid a possible psychological effect, the subjects should taste the drinks blind (i.e., they don't know the identity of the drink). Therefore, we reject the null hypothesis. A point estimate for the difference in two population means is simply the difference in the corresponding sample means. Then, under the H0, $$ \frac { \bar { B } -\bar { A } }{ S\sqrt { \frac { 1 }{ m } +\frac { 1 }{ n } } } \sim { t }_{ m+n-2 } $$, $$ \begin{align*} { S }_{ A }^{ 2 } & =\frac { \left\{ 59520-{ \left( 10\ast { 75 }^{ 2 } \right) } \right\} }{ 9 } =363.33 \\ { S }_{ B }^{ 2 } & =\frac { \left\{ 56430-{ \left( 10\ast { 72}^{ 2 } \right) } \right\} }{ 9 } =510 \\ \end{align*} $$, $$ S^p_2 =\cfrac {(9 * 363.33 + 9 * 510)}{(10 + 10 -2)} = 436.665 $$, $$ \text{the test statistic} =\cfrac {(75 -72)}{ \left\{ \sqrt{439.665} * \sqrt{ \left(\frac {1}{10} + \frac {1}{10}\right)} \right\} }= 0.3210 $$. Hypothesis test. We find the critical T-value using the same simulation we used in Estimating a Population Mean.. The two types of samples require a different theory to construct a confidence interval and develop a hypothesis test. 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: Figure \(\PageIndex{1}\) illustrates the conceptual framework of our investigation in this and the next section. 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. 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. For instance, they might want to know whether the average returns for two subsidiaries of a given company exhibit a significant difference. The results of such a test may then inform decisions regarding resource allocation or the rewarding of directors. We want to compare the gas mileage of two brands of gasoline. 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. The two populations are independent. Now we can apply all we learned for the one sample mean to the difference (Cool!). Use these data to produce a point estimate for the mean difference in the hotel rates for the two cities. The number of observations in the first sample is 15 and 12 in the second sample. If this rule of thumb is satisfied, we can assume the variances are equal. where \(D_0\) is a number that is deduced from the statement of the situation. If we find the difference as the concentration of the bottom water minus the concentration of the surface water, then null and alternative hypotheses are: \(H_0\colon \mu_d=0\) vs \(H_a\colon \mu_d>0\). 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. A difference between the two samples depends on both the means and the standard deviations. 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. 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. Biometrika, 29(3/4), 350. doi:10.2307/2332010 Thus, \[(\bar{x_1}-\bar{x_2})\pm z_{\alpha /2}\sqrt{\frac{s_{1}^{2}}{n_1}+\frac{s_{2}^{2}}{n_2}}=0.27\pm 2.576\sqrt{\frac{0.51^{2}}{174}+\frac{0.52^{2}}{355}}=0.27\pm 0.12 \nonumber \]. For a right-tailed test, the rejection region is \(t^*>1.8331\). This assumption does not seem to be violated. Given data from two samples, we can do a signficance test to compare the sample means with a test statistic and p-value, and determine if there is enough evidence to suggest a difference between the two population means. The conditions for using this two-sample T-interval are the same as the conditions for using the two-sample T-test. Therefore, $$ { t }_{ { n }_{ 1 }+{ n }_{ 2 }-2 }=\frac { { \bar { x } }_{ 1 }-{ \bar { x } }_{ 2 } }{ { S }_{ p }\sqrt { \left( \frac { 1 }{ { n }_{ 1 } } +\frac { 1 }{ { n }_{ 2 } } \right) } } $$. 9.1: Prelude to Hypothesis Testing with Two Samples, 9.3: Inferences for Two Population Means - Unknown Standard Deviations, \(100(1-\alpha )\%\) Confidence Interval for the Difference Between Two Population Means: Large, Independent Samples, Standardized Test Statistic for Hypothesis Tests Concerning the Difference Between Two Population Means: Large, Independent Samples, status page at https://status.libretexts.org. The mean glycosylated hemoglobin for the whole study population was 8.971.87. It seems natural to estimate \(\sigma_1\) by \(s_1\) and \(\sigma_2\) by \(s_2\). The critical T-value comes from the T-model, just as it did in Estimating a Population Mean. Again, this value depends on the degrees of freedom (df). 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 . Refer to Questions 1 & 2 and use 19.48 as the degrees of freedom. The only difference is in the formula for the standardized test statistic. Interpret the confidence interval in context. The parameter of interest is \(\mu_d\). 2. Which method [] The same process for the hypothesis test for one mean can be applied. All received tutoring in arithmetic skills. We either give the df or use technology to find the df. Our test statistic, -3.3978, is in our rejection region, therefore, we reject the null hypothesis. The difference makes sense too! 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: \(t^*=\dfrac{\bar{d}-0}{\frac{s_d}{\sqrt{n}}}\). Given this, there are two options for estimating the variances for the independent samples: When to use which? 734) of the t-distribution with 18 degrees of freedom. 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. Here are some of the results: https://assess.lumenlearning.com/practice/10bbd676-7ed8-476f-897b-43ac6076b4d2. 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. We can thus proceed with the pooled t-test. With a significance level of 5%, there is enough evidence in the data to suggest that the bottom water has higher concentrations of zinc than the surface level. Carry out a 5% test to determine if the patients on the special diet have a lower weight. The 95% confidence interval for the mean difference, \(\mu_d\) is: \(\bar{d}\pm t_{\alpha/2}\dfrac{s_d}{\sqrt{n}}\), \(0.0804\pm 2.2622\left( \dfrac{0.0523}{\sqrt{10}}\right)\). Note that these hypotheses constitute a two-tailed test. In order to test whether there is a difference between population means, we are going to make three assumptions: The two populations have the same variance. When considering the sample mean, there were two parameters we had to consider, \(\mu\) the population mean, and \(\sigma\) the population standard deviation. That is, \(p\)-value=\(0.0000\) to four decimal places. The critical value is the value \(a\) such that \(P(T>a)=0.05\). We demonstrate how to find this interval using Minitab after presenting the hypothesis test. The sample mean difference is \(\bar{d}=0.0804\) and the standard deviation is \(s_d=0.0523\). Confidence Interval to Estimate 1 2 Does the data suggest that the true average concentration in the bottom water is different than that of surface water? Perform the test of Example \(\PageIndex{2}\) using the \(p\)-value approach. (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.). 40 views, 2 likes, 3 loves, 48 comments, 2 shares, Facebook Watch Videos from Mt Olive Baptist Church: Worship We assume that 2 1 = 2 1 = 2 1 2 = 1 2 = 2 H0: 1 - 2 = 0 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. 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. Figure \(\PageIndex{1}\) illustrates the conceptual framework of our investigation in this and the next section. 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 \(2\). 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? There is no indication that there is a violation of the normal assumption for both samples. 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 \(t^*=\dfrac{\bar{x}_1-\bar{x}_2-0}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\). The 99% confidence interval is (-2.013, -0.167). 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. The samples must be independent, and each sample must be large: \(n_1\geq 30\) and \(n_2\geq 30\). Instructions : Use this T-Test Calculator for two Independent Means calculator to conduct a t-test for two population means ( \mu_1 1 and \mu_2 2 ), with unknown population standard deviations. Therefore, we are in the paired data setting. 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. H 1: 1 2 There is a difference between the two population means. 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 . follows a t-distribution with \(n_1+n_2-2\) degrees of freedom. We randomly select 20 males and 20 females and compare the average time they spend watching TV. Using the Central Limit Theorem, if the population is not normal, then with a large sample, the sampling distribution is approximately normal. Independent Samples Confidence Interval Calculator. If the two are equal, the ratio would be 1, i.e. 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. The populations are normally distributed or each sample size is at least 30. The null hypothesis is that there is no difference in the two population means, i.e. 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). In the context of the problem we say we are \(99\%\) confident that the average level of customer satisfaction for Company \(1\) is between \(0.15\) and \(0.39\) points higher, on this five-point scale, than that for Company \(2\). Construct a 95% confidence interval for 1 2. That is, you proceed with the p-value approach or critical value approach in the same exact way. When dealing with large samples, we can use S2 to estimate 2. 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 \(2\). To learn how to construct a confidence interval for the difference in the means of two distinct populations using large, independent samples. 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. 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. Minitab will calculate the confidence interval and a hypothesis test simultaneously. 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. This is a two-sided test so alpha is split into two sides. (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.). We should proceed with caution. 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. Create a relative frequency polygon that displays the distribution of each population on the same graph. Do the data provide sufficient evidence to conclude that, on the average, the new machine packs faster? Putting all this together gives us the following formula for the two-sample T-interval. An obvious next question is how much larger? This test apply when you have two-independent samples, and the population standard deviations \sigma_1 1 and \sigma_2 2 and not known. 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 can proceed with using our tools, but we should proceed with caution. However, since these are samples and therefore involve error, we cannot expect the ratio to be exactly 1. \(t^*=\dfrac{\bar{x}_1-\bar{x_2}-0}{\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}}\), will have a t-distribution with degrees of freedom, \(df=\dfrac{(n_1-1)(n_2-1)}{(n_2-1)C^2+(1-C)^2(n_1-1)}\). From an international perspective, the difference in US median and mean wealth per adult is over 600%. Is this an independent sample or paired sample? Ten pairs of data were taken measuring zinc concentration in bottom water and surface water (zinc_conc.txt). 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 \(2\). Without reference to the first sample we draw a sample from Population \(2\) and label its sample statistics with the subscript \(2\). Question: Confidence interval for the difference between the two population means. (The actual value is approximately \(0.000000007\).). The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We are interested in the difference between the two population means for the two methods. Agreement was assessed using Bland Altman (BA) analysis with 95% limits of agreement. 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 \(2\). Therefore, if checking normality in the populations is impossible, then we look at the distribution in the samples. We are 95% confident that the difference between the mean GPA of sophomores and juniors is between -0.45 and 0.173. Samples from two distinct populations are independent if each one is drawn without reference to the other, and has no connection with the other. 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}. Look at the distribution in the same process for the two samples are large we can the! 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