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?

\(H_0: \mu = \mu_0\)
\(H_a: \mu \ne \mu_0\) OR \(H_a: \mu > \mu_0\) OR \(H_a: \mu < \mu_0\)
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 \(H_a: p > p_0\) OR \(H_a: p < p_0\)_{ }
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?

\(H_0: \mu_D = 0\)
\(H_a: \mu_D \ne 0\) 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 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?

\(H_0: p_1 = p_2\)
\(H_a: p_1 \ne p_2 \) 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 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 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? 
\(H_0: \rho = 0\)
\(H_a: \rho \ne 0\)
\(t=\frac{r\sqrt{n2}}{\sqrt{1r^2}}\)

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
