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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

(printable version )

Summary Table for Statistical Techniques
 
Inference
Parameter
Statistic
Type of Data
Examples
Analysis
Minitab Command
Conditions
1 Estimating a Mean

One Population Mean

μ

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

μ

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?

Ho: μ = μ0

Ha: μ ≠ μ0
OR
Ha: μ > μ0
OR
Ha: μ < μ0

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

π

Sample Proportion

\(\hat{\pi}\)

Categorical (Binary)

What is the proportion of males in the world?

What is the proportion of students that smoke?

1-proportion Z-interval

\(\hat{\pi }\pm z_{\alpha /2}\sqrt{\frac{\hat{\pi }\cdot \left ( 1-\hat{\pi } \right )}{n}}\)

Stat > Basic statistics > 1-sample proportion

have at least 5 in each category

4 Test About a Proportion

One Population Proportion

π

Sample Proportion \(\hat{\pi}\) Categorical (Binary)

Is the proportion of females different from 0.5?

Is the proportion of students who fail STAT 500 less than 0.1?

Ho: π = π0

Ha: π ≠ π0
OR
Ha: π > π0
OR
Ha: π < π0

The one proportion Z-test:

\(z=\frac{\hat{\pi }-\pi _{0}}{\sqrt{\frac{\pi _{0}\left ( 1- \pi _{0}\right )}{n}}}\)

Stat > Basic statistics > 1-sample proportion

nπ0 ≥ 5 and n (1 - π0) ≥ 5

5 Estimating the Difference of Two Means

Difference in two population means

μ1 - μ2

Difference in two sample means

\(\bar{y}_{1} - \bar{y}_{2}\)

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

\(\bar{y}_{1}-\bar{y}_{2}\pm t_{\alpha /2}\cdot s.e.\left (\bar{y}_{1}-\bar{y}_{2}  \right )\)

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

μ1 - μ2

Difference in two sample means

\(\bar{y}_{1} - \bar{y}_{2}\)

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?

Ho: μ1 = μ2

Ha: μ1 ≠ μ2 OR Ho: μ1 > μ2 OR Ha: μ1 < μ2

The 2-sample t-test:

\(t=\frac{\left (\bar{y}_{1}-\bar{y}_{2}  \right )-0}{s.e.\left (\bar{y}_{1}-\bar{y}_{2}  \right )} \)

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

μD

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

μD

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?

Ho: μD = 0

Ha: μD ≠ 0
OR
Ho: μD > 0
OR
Ha: μD < 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

π1 - π2

Difference in two sample proportions

\(\hat{\pi }_{1} - \hat{\pi }_{2}\)

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

\(\hat{\pi _{1}}-\hat{\pi _{2}}\pm z_{\alpha /2}\cdot s.e.\left ( \hat{\pi _{1}}-\hat{\pi _{2}} \right )\)

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

π1 - π2

Difference in two sample proportions

\(\hat{\pi }_{1} - \hat{\pi }_{2}\)

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?

Ho: π1 = π2

Ha: π1 ≠ π2
OR
Ha: π1 > π2
OR
Ha: π1 < π2

The two proportion z test:

\(z=\frac{\hat{\pi }_{1}-\hat{\pi }_{2}}{\sqrt{\hat{\pi }\left ( 1-\hat{\pi } \right )\left ( \frac{1}{n_{1}}+ \frac{1}{n_{2}}\right )}}\)

\(\hat{\pi}=\frac{y_{1}+y_{2}}{n_{1}+n_{2}}\)

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

β1

Sample estimate of the slope

b1

Numerical

Is there a linear relationship between height and weight of a person?

Ho: β1 = 0

Ha: β1 ≠ 0 OR Ha: β1 > 0 OR Ha: β1 < 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

μ1, μ2, ... , μt

Sample means of the t populations

x1, x2, ... , xt

Numerical

Is there a difference between the mean GPA of freshman, sophomore, junior, and senior classes?

Ho: μ1 = μ2 = ... = μt

Ha: 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 to Compare Two Population Variances

Population variances of 2 populations

σ12, σ22

Sample variances of 2 populations

s12, s22

Numerical

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

Ho: σ12 = σ22

Ha: σ12 ≠ σ22

\(F=\frac{s_{1}^{2}}{s_{2}^{2}}\)

Stat > ANOVA > Test of homogeneity

Each population is normally distributed

Independent samples from the 2 populations