12.1  Summary of Statistical Techniques
12.1  Summary of Statistical TechniquesTabbed Flow Charts
Summary Table for Statistical Techniques
Estimate a Value
 Estimating a Mean
 Estimating a Proportion
 Estimating the difference of two means
 Estimating a mean with paired data
 Estimating the difference of two proportions
Test a hypothesis
 Test about a mean
 Test about a proportion
 Test to compare two means (independent)
 Test to compare two means (paired)
 Test to compare two proportions
 Test about a slope
 Test to compare several means
 Test of Strength & Direction of Linear Relationship of 2 Quantitative Variables
 Test to Compare Two Population Variances
Estimating a Mean
Parameter
One opulation mean, \(\mu\)
Statistic
Sample mean, \(\bar{x}\)
Type of Data
Numerical
Analysis
1sample tinterval
\(\bar{x}\pm t_{\alpha /2}\cdot \frac{s}{\sqrt{n}}\)
Minitab Command
Stat > Basic statistics > 1sample t
Conditions
data approximately normal OR
have a large sample size (n ≥ 30)
Examples
 What is the average weight of adults?
 What is the average cholesterol level of adult females?
Test About a Mean
Parameter
One population mean, \(\mu\)
Statistic
Sample mean, \(\bar{x}\)
Type of Data
Numerical
Analysis
\(H_0\colon \mu = \mu_0\)
\(H_a\colon \mu \ne \mu_0\) OR
\(H_a\colon \mu > \mu_0\) OR
\(H_a\colon \mu < \mu_0\)
1sample ttest:
\(t=\frac{\bar{x}\mu_{0}}{\frac{s}{\sqrt{n}}}\)
Minitab Command
Stat > Basic statistics > 1sample t
Conditions
data approximately normal
OR
have a large sample size (n ≥ 30)
Examples
 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?
Estimating a Proportion
Parameter
One population proportion \(p\)
Statistic
Sample proportion, \(\hat{p}\)
Type of Data
Categorical (Binary)
Analysis
1proportion Zinterval:
\( \hat{p}\pm z_{\alpha /2}\sqrt{\frac{\hat{p}\cdot \left ( 1\hat{p} \right )}{n}}\)
Minitab Command
Stat > Basic statistics > 1sample proportion
Conditions
have at least 5 in each categoryExamples
 What is the proportion of males in the world?
 What is the proportion of students that smoke?
Test about a Proportion
Parameter
One population proportion, \(p\)
Statistic
Sample proportion, \(\hat{p}\)
Type of Data
Categorical (Binary)
Analysis
\(H_0\colon p = p_0\)
\(H_a\colon p \ne p_0\) OR
\(H_a\colon p > p_0\) OR
\(H_a\colon p < p_0\)
1proportion Ztest:
\(z=\frac{\hat{p}p _{0}}{\sqrt{\frac{p _{0}\left ( 1 p _{0}\right )}{n}}}\)
Minitab Command
Stat > Basic statistics > 1sample proportion
Conditions
\(np_0 \geq 5\) and
\(n (1  p_0) \geq 5\)
Examples
 Is the proportion of females different from 0.5?
 Is the proportion of students who fail STAT 500 less than 0.1?
Estimating the Difference of Two Means*
Parameter
Difference in two population means,
\(\mu_1  \mu_2\)
Statistic
Difference in two sample means,
\(\bar{x}_{1}  \bar{x}_{2}\)
Type of Data
Numerical
Analysis
2sample tinterval:
\(\bar{x}_{1}\bar{x}_{2}\pm t_{\alpha /2}\cdot \\\hat{s.e.}\left (\bar{x}_{1}\bar{x}_{2} \right )\)
Minitab Command
Stat > Basic statistics > 2sample t
Conditions
Independent samples from the two populations
Data in each sample are about normal or large samples
Examples
 How different are the mean GPAs of males and females?
 How many fewer colds do vitamin C takers get, on average, than nonvitamin takers?
Test to Compare Two Means*
Parameter
Difference in two population means,
\(\mu_1  \mu_2\)
Statistic
Difference in two sample means,
\(\bar{x}_{1}  \bar{x}_{2}\)
Type of Data
Numerical
Analysis
\(H_0\colon \mu_1 = \mu_2\) \(H_a\colon \mu_1 \ne \mu_2\) OR
\(H_a\colon \mu_1 > \mu_2\) OR
\(H_a\colon \mu_1 < \mu_2\)
2sample ttest: \(t=\frac{\left (\bar{x}_{1}\bar{x}_{2} \right )0}{\hat{s.e.}\left (\bar{x}_{1}\bar{x}_{2} \right )} \)
Minitab Command
Stat > Basic statistics > 2sample t
Conditions
Independent samples from the two populations
Data in each sample are about normal or large samples
Examples
 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?
*(The Standard Error (S.E.) will depend on pooled vs unpooled)
Estimating a Mean with Paired Data
Parameter
Mean of paired difference,
\(\mu_D\)
Statistic
Sample mean of difference,
\(\bar{d}\)
Type of Data
Numerical
Analysis
paired tinterval:
\(\bar{d}\pm t_{\alpha /2}\cdot \frac{s_{d}}{\sqrt{n}}\)
Minitab Command
Stat > Basic statistics > Paired t
Conditions
Differences approximately normal OR
Have a large number of pairs (n ≥ 30)
Examples
 What is the difference in pulse rates, on the average, before and after exercise?
Test about a Mean with Paired Data
Parameter
Mean of paired difference,
\(\mu_D\)
Statistic
Sample mean of difference,
\(\bar{d}\)
Type of Data
Numerical
Analysis
\(H_0\colon \mu_D = 0\)
\(H_a\colon \mu_D \ne 0\) OR
\(H_a\colon \mu_D > 0\) OR
\(H_a\colon \mu_D < 0\)
ttest statistic:
\(t=\frac{\bar{d}0}{\frac{s_d}{\sqrt{n}}}\)
Minitab Command
Stat > Basic statistics > Paired t
Conditions
Differences approximately normal OR
Have a large number of pairs (n ≥ 30)
Examples
 Is the difference in IQ of pairs of twins zero?
 Are the pulse rates of people higher after exercise?
Estimating the Difference of Two Proportions
Parameter
Difference in two population proportions,
\(p_1  p_2\)
Statistic
Difference in two sample proportions,
\(\hat{p}_{1}  \hat{p}_{2}\)
Type of Data
Categorical (Binary)
Analysis
2proportions Zinterval:
\(\hat{p} _{1}\hat{p} _{2}\pm z_{\alpha /2}\cdot\\ \hat{s.e.}\left ( \hat{p} _{1}\hat{p} _{2} \right )\)
Minitab Command
Stat > Basic statistics > 2 proportions
Conditions
Independent samples from the two populations
Have at least 5 in each category for both populations
Examples
 How different are the percentages of male and female smokers?
 How different are the percentages of upper and lowerclass binge drinkers?
Test to Compare Two Proportions
Parameter
Difference in two population proportions,
\(p_1  p_2\)
Statistic
Difference in two sample proportions,
\(\hat{p}_{1}  \hat{p}_{2}\)
Type of Data
Categorical (Binary)
Analysis
\(H_0\colon p_1 = p_2\)
\(H_a\colon p_1 \ne p_2 \) OR
\(H_a\colon p_1 > p_2\) OR
\(H_a\colon p_1 < p_2\)
2proportion Ztest:
\(z^*=\frac{\hat{p}_{1}\hat{p}_{2}}{\sqrt{\hat{p}^*\left ( 1\hat{p}^* \right )\left ( \frac{1}{n_{1}}+ \frac{1}{n_{2}}\right )}}\)
\(\hat{p}^*=\dfrac{x_{1}+x_{2}}{n_{1}+n_{2}}\)
Minitab Command
Stat > Basic statistics > 2 proportions
Conditions
Independent samples from the two populations
Have at least 5 in each category for both populations
Examples

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?
Relationship in a 2Way Table
Parameter
Relationship between two categorical variables, OR
difference in two or more population proportions
Statistic
The observed counts in a twoway table
Type of Data
Categorical
Analysis
\(H_0\colon\text{The two variables are not related}\)
\(H_a\colon\text{The two variables are related}\)
Chisquare test statistic:
\(X^2=\sum_{\text{all cells}}\frac{(\text{ObservedExpected})^2}{\text{Expected}}\)
Minitab Command
Stat > Tables > Chi square Test for Association
Conditions
All expected counts should be greater than 1
At least 80% of the cells should have an expected count greater than 5
Examples
 Is there a relationship between smoking and lung cancer?
 Do the proportions of students in each class who smoke differ?
Test About a Slope
Parameter
Slope of the population regression line,
\(\beta_1\)
Statistic
Sample estimate of the slope,
\(b_1\)
Type of Data
Numerical
Analysis
\(H_0\colon \beta_1 = 0\)
\(H_a\colon \beta_1 \ne 0\) OR
\(H_a\colon \beta_1 > 0\) OR
\(H_a\colon \beta_1 < 0\)
ttest with n  2 degrees of freedom:
\(t=\dfrac{b_{1}0}{\hat{s.e.}\left ( b_{1} \right )}\)
Minitab Command
Stat > Regression > Regression
Conditions
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
Examples

Is there a linear relationship between height and weight of a person?
Test to Compare Several Means
Parameter
Population means of the t populations,
\(\mu_1, \mu_2, \cdots , \mu_t\)
Statistic
Sample means of the t populations,
\(x_1, x_2, \cdots , x_t\)
Type of Data
Numerical
Analysis
\(H_0\colon \mu_1 = \mu_2 = ... = \mu_t\)
\(H_a\colon \text{not all the means are equal}\)
Ftest for oneway ANOVA:
\(F=\dfrac{MST}{MSE}\)
Minitab Command
Stat > ANOVA > Oneway
Conditions
Each population is normally distributed
Independent samples from the t populations
Equal population standard deviations
Examples

Is there a difference between the mean GPA of freshman, sophomore, junior, and senior classes?
Test of Strength & Direction of Linear Relationship of 2 Quantitative Variables
Parameter
Population correlation,
\(\rho\)
"rho"
Statistic
Sample correlation,
\(r\)
Type of Data
Numerical
Analysis
\(H_0\colon \rho = 0\)
\(H_a\colon \rho \ne 0\)
ttest statistic:
\(t=\frac{r\sqrt{n2}}{\sqrt{1r^2}}\)
Minitab Command
Stat > Basic Statistics > Correlation
Conditions
2 variables are continuous
Related pairs
No significant outliers
Normality of both variables
Linear relationship between the variables
Examples

Is there a linear relationship between height and weight?
Test to Compare Two Population Variances
Parameter
Population variances of two populations,
\(\sigma_{1}^{2}, \sigma_{2}^{2}\)
Statistic
Sample variances of two populations,
\(s_{1}^{2}, s_{2}^{2}\)
Type of Data
Numerical
Analysis
\(H_0\colon \sigma_{1}^{2} = \sigma_{2}^{2}\)
\(H_2\colon \sigma_{1}^{2} \ne \sigma_{2}^{2}\)
Ftest statistic:
\(F=\frac{s_{1}^{2}}{s_{2}^{2}}\)
Minitab Command
Stat > Basic statistics > 2 variances
Conditions
Each population is normally distributed
Independent samples from the 2 populations
Examples

Are the variances of length of lumber produced by Company A different from those produced by Company B?