12.1  Summary Table for Statistical Techniques
Unit Summary 

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? 
1sample tinterval \(\bar{x}\pm t_{\alpha /2}\cdot \frac{s}{\sqrt{n}}\) 
Stat > Basic statistics > 1sample 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? 
\(H_0: \mu = \mu_0\) \(H_a: \mu \ne \mu_0\) The 1sample ttest: \(t=\frac{\bar{x}\mu_{0}}{\frac{s}{\sqrt{n}}}\) 
Stat > Basic statistics > 1sample 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? 
1proportion Zinterval \(\hat{p}\pm z_{\alpha /2}\sqrt{\frac{\hat{p}\cdot \left ( 1\hat{p} \right )}{n}}\) 
Stat > Basic statistics > 1sample 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 The one proportion Ztest: \(z=\frac{\hat{p}p _{0}}{\sqrt{\frac{p _{0}\left ( 1 p _{0}\right )}{n}}}\) 
Stat > Basic statistics > 1sample 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 nonvitamin takers? 
2sample tinterval \(\bar{x}_{1}\bar{x}_{2}\pm t_{\alpha /2}\cdot s.e.\left (\bar{x}_{1}\bar{x}_{2} \right )\) 
Stat > Basic statistics > 2sample 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 nonexercisers 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 2sample ttest: \(t=\frac{\left (\bar{x}_{1}\bar{x}_{2} \right )0}{s.e.\left (\bar{x}_{1}\bar{x}_{2} \right )} \) 
Stat > Basic statistics > 2sample 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 tinterval \(\bar{d}\pm t_{\alpha /2}\cdot \frac{s_{d}}{\sqrt{n}}\) 
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? 
\(H_0: \mu_D = 0\) \(H_a: \mu_D \ne 0\) \(t=\frac{\bar{d}0}{\frac{s_{d}}{\sqrt{n}}}\) 
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 lowerclass binge drinkers? 
twoproportions Zinterval \(\hat{p _{1}}\hat{p _{2}}\pm z_{\alpha /2}\cdot s.e.\left ( \hat{p _{1}}\hat{p _{2}} \right )\) 
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? 
\(H_0: p_1 = p_2\) \(H_a: p_1 \ne p_2 \) 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 2Way Table  Relationship between two categorical variables or difference in two or more population proportions  The observed counts in a twoway table  Categorical 
Is there a relationship between smoking and lung cancer? Do the proportions of students in each class who smoke differ? 
H_{o}: The two variables are not related H_{a}: The two variables are related The chisquare statistic: \(X^2=\sum_{\text{all cells}}\frac{(\text{ObservedExpected})^2}{\text{Expected}}\) 
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 b_{1} 
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 ttest 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 Ftest for oneway 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 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 