# 12.1 - Summary Table for Statistical Techniques

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 Unit Summary Review for the Statistical Techniques We Have Learned Summary Table for Statistical Techniques

### Review for the Statistical Techniques We Have Learned

We have learned many different formula and techniques to analyze different types of problems in this course. It is easier to know what technique to apply when we are only talking about certain topics. In real life and in the final exam, we don't have that hint and it is most important to know when to use what statistical technique. The following summary table for statistical techniques provides a review for the subjects we have learned in this course. It is also a good reference when you work on the next section -- to choose the statistical techniques for the given problem.

### Summary Table for Statistical Techniques

 Summary Table for Statistical Techniques Inference Parameter Statistic Type of Data Examples Analysis Minitab Command Conditions 1 Estimating a Mean One Population Mean $$\mu$$ Sample Mean $$\bar{x}$$ Numerical What is the average weight of adults? What is the average cholesterol level of adult females? 1-sample t-interval $$\bar{x}\pm t_{\alpha /2}\cdot \frac{s}{\sqrt{n}}$$ Stat > Basic statistics > 1-sample t data approximately normal OR have a large sample size (n ≥ 30) 2 Test About a Mean One population Mean $$\mu$$ Sample Mean $$\bar{x}$$ Numerical Is the average GPA of juniors at Penn State higher than 3.0? Is the average winter temperature in State College less than 42°F? $$H_0: \mu = \mu_0$$ $$H_a: \mu \ne \mu_0$$ OR$$H_a: \mu > \mu_0$$ OR$$H_a: \mu < \mu_0$$ The 1-sample t-test: $$t=\frac{\bar{x}-\mu_{0}}{\frac{s}{\sqrt{n}}}$$ Stat > Basic statistics > 1-sample t data approximately normal OR have a large sample size (n ≥ 30) 3 Estimating a Proportion One Population Proportion $$p$$ Sample Proportion $$\hat{p}$$ Categorical (Binary) What is the proportion of males in the world? What is the proportion of students that smoke? 1-proportion Z-interval $$\hat{p}\pm z_{\alpha /2}\sqrt{\frac{\hat{p}\cdot \left ( 1-\hat{p} \right )}{n}}$$ Stat > Basic statistics > 1-sample proportion have at least 5 in each category 4 Test About a Proportion One Population Proportion $$p$$ Sample Proportion $$\hat{p}$$ Categorical (Binary) Is the proportion of females different from 0.5? Is the proportion of students who fail STAT 500 less than 0.1? $$H_0: p = p_0$$ $$H_a: p \ne p_0$$ OR$$H_a: p > p_0$$ OR $$H_a: p < p_0$$ The one proportion Z-test: $$z=\frac{\hat{p}-p _{0}}{\sqrt{\frac{p _{0}\left ( 1- p _{0}\right )}{n}}}$$ Stat > Basic statistics > 1-sample proportion $$np_0 \geq 5$$ and $$n (1 - p_0) \geq 5$$ 5 Estimating the Difference of Two Means Difference in two population means $$\mu_1 - \mu_2$$ Difference in two sample means $$\bar{x}_{1} - \bar{x}_{2}$$ Numerical How different are the mean GPAs of males and females? How many fewer colds do vitamin C takers get, on average, than non-vitamin takers? 2-sample t-interval $$\bar{x}_{1}-\bar{x}_{2}\pm t_{\alpha /2}\cdot s.e.\left (\bar{x}_{1}-\bar{x}_{2} \right )$$ Stat > Basic statistics > 2-sample t Independent samples from the two populations Data in each sample are about normal or large samples 6 Test to Compare Two Means Difference in two population means $$\mu_1 - \mu_2$$ Difference in two sample means $$\bar{x}_{1} - \bar{x}_{2}$$ Numerical Do the mean pulse rates of exercisers and non-exercisers differ? Is the mean EDS score for dropouts greater than the mean EDS score for graduates? $$H_0: \mu_1 = \mu_2$$ $$H_a: \mu_1 \ne \mu_2$$ OR $$H_a: \mu_1 > \mu_2$$ OR $$H_a: \mu_1 < \mu_1$$ The 2-sample t-test: $$t=\frac{\left (\bar{x}_{1}-\bar{x}_{2} \right )-0}{s.e.\left (\bar{x}_{1}-\bar{x}_{2} \right )}$$ Stat > Basic statistics > 2-sample t Independent samples from the two populations Data in each sample are about normal or large samples 7 Estimating a Mean with Paired Data Mean of paired difference $$\mu_D$$ Sample mean of difference $$\bar{d}$$ Numerical What is the difference in pulse rates, on the average, before and after exercise? paired t-interval $$\bar{d}\pm t_{\alpha /2}\cdot \frac{s_{d}}{\sqrt{n}}$$ Stat > Basic statistics > Paired t Differences approximately normal OR Have a large number of pairs (n ≥ 30) 8 Test About a Mean with Paired Data Mean of paired difference $$\mu_D$$ Sample mean of difference $$\bar{d}$$ Numerical Is the difference in IQ of pairs of twins zero? Are the pulse rates of people higher after exercise? $$H_0: \mu_D = 0$$ $$H_a: \mu_D \ne 0$$ OR $$H_a: \mu_D > 0$$ OR $$H_a: \mu_D < 0$$ $$t=\frac{\bar{d}-0}{\frac{s_{d}}{\sqrt{n}}}$$ Stat > Basic statistics > Paired t Differences approximately normal OR Have a large number of pairs (n ≥ 30) 9 Estimating the Difference of Two Proportions Difference in two population proportions $$p_1 - p_2$$ Difference in two sample proportions $$\hat{p}_{1} - \hat{p}_{2}$$ Categorical (Binary) How different are the percentages of male and female smokers? How different are the percentages of upper- and lower-class binge drinkers? two-proportions Z-interval $$\hat{p _{1}}-\hat{p _{2}}\pm z_{\alpha /2}\cdot s.e.\left ( \hat{p _{1}}-\hat{p _{2}} \right )$$ Stat > Basic statistics > 2 proportions Independent samples from the two populations Have at least 5 in each category for both populations 10 Test to Compare Two Proportions Difference in two population proportions $$p_1 - p_2$$ Difference in two sample proportions $$\hat{p}_{1} - \hat{p}_{2}$$ Categorical (Binary) Is the percentage of males with lung cancer higher than the percentage of females with lung cancer? Are the percentages of upper- and lower- class binge drinkers different? $$H_0: p_1 = p_2$$ $$H_a: p_1 \ne p_2$$ OR $$H_a: p_1 > p_2$$ OR $$H_a: p_1 < p_2$$ The two proportion z test: $$z=\frac{\hat{p}_{1}-\hat{p}_{2}}{\sqrt{\hat{p}\left ( 1-\hat{p} \right )\left ( \frac{1}{n_{1}}+ \frac{1}{n_{2}}\right )}}$$ $$\hat{p}=\frac{x_{1}+x_{2}}{n_{1}+n_{2}}$$ Stat > Basic statistics > 2 proportions Independent samples from the two populations Have at least 5 in each category for both populations 11 Relationship in a 2-Way Table Relationship between two categorical variables or difference in two or more population proportions The observed counts in a two-way table Categorical Is there a relationship between smoking and lung cancer? Do the proportions of students in each class who smoke differ? Ho: The two variables are not related Ha: The two variables are related The chi-square statistic: $$X^2=\sum_{\text{all cells}}\frac{(\text{Observed-Expected})^2}{\text{Expected}}$$ Stat > Tables > Chi square Test All expected counts should be greater than 1 At least 80% of the cells should have an expected count greater than 5 12 Test About a Slope Slope of the population regression line $$\beta_1$$ Sample estimate of the slope b1 Numerical Is there a linear relationship between height and weight of a person? $$H_0: \beta_1 = 0$$ $$H_a: \beta_1 \ne 0$$ OR $$H_a: \beta_1 > 0$$ OR $$H_a: \beta_1 < 0$$ The t-test with n - 2 degrees of freedom: $$t=\frac{b_{1}-0}{s.e.\left ( b_{1} \right )}$$ Stat > Regression > Regression The form of the equation that links the two variables must be correct The error terms are normally distributed The errors terms have equal variances The error terms are independent of each other 13 Test to Compare Several Means Population means of the t populations $$\mu_1, \mu_2, \cdots , \mu_t$$ Sample means of the t populations $$x_1, x_2, \cdots , x_t$$ Numerical Is there a difference between the mean GPA of freshman, sophomore, junior, and senior classes? $$H_0: \mu_1 = \mu_2 = ... = \mu_t$$ $$H_a: \text{not all the means are equal}$$ The F-test for one-way ANOVA: $$F=\frac{MSTR}{MSE}$$ Stat > ANOVA > Oneway Each population is normally distributed Independent samples from the t populations Equal population standard deviations 14 Test of Strength & Direction of Linear Relationship of 2 Quantitative Variables Population Correlation $$\rho$$ "rho" Sample correlation $$r$$ Numerical Is there a linear relationship between height and weight? $$H_0: \rho = 0$$ $$H_a: \rho \ne 0$$ $$t=\frac{r\sqrt{n-2}}{\sqrt{1-r^2}}$$ Stat > Basic Statistics > Correlation 2 variables are continuous Related pairs No significant outliers Normality of both variables Linear relationship between the variables 15 Test to Compare Two Population Variances Population variances of 2 populations $$\sigma_{1}^{2}, \sigma_{2}^{2}$$ Sample variances of 2 populations $$s_{1}^{2}, s_{2}^{2}$$ Numerical Are the variances of length of lumber produced by Company A different from those produced by Company B $$H_0: \sigma_{1}^{2} = \sigma_{2}^{2}$$ $$H_2: \sigma_{1}^{2} \ne \sigma_{2}^{2}$$ $$F=\frac{s_{1}^{2}}{s_{2}^{2}}$$ Stat > Basic statistics > 2 variances Each population is normally distributed Independent samples from the 2 populations