# Hypothesis Testing with Two Samples

## Two Population Means with Unknown Standard Deviations

1. The two independent samples are simple random samples from two distinct populations.
2. For the two distinct populations



The comparison of two population means is very common. A difference between the two samples depends on both the means and the standard deviations. Very different means can occur by chance if there is great variation among the individual samples. To account for the variation, we take the difference of the sample means,


 
  , and divide by the standard error to standardize the difference. The result is a t-score test statistic.
Because we do not know the population standard deviations, we estimate them using the two sample standard deviations from our independent samples. For the hypothesis test, we calculate the estimated standard deviation, or standard error, of the difference in sample means,

 
  
The standard error is calculated as follows:
The test statistic (t-score) is calculated as follows:
  


The number of degrees of freedom ( requires a somewhat complicated calculation. However, a computer or calculator calculates it easily. The df are not always a whole number. The test statistic calculated previously is approximated by the Student’s t-distribution with df as follows:
Degrees of freedom
When both sample sizes n1 and n2 are five or larger, the Student’s t approximation is very good. Notice that the sample variances (s1)2 and (s2)2 are not pooled. (If the question comes up, do not pool the variances.)


Cohen’s Standards for Small, Medium, and Large Effect SizesCohen’s  is a measure of effect size based on the differences between two means. Cohen’s d, named for U.S. statistician Jacob Cohen, measures the relative strength of the differences between the means of two populations based on sample data. The calculated value of effect size is then compared to Cohen’s standards of small, medium, and large effect sizes.
Cohen’s d is the measure of the difference between two means divided by the pooled standard deviation: 
 where 
.

### References

Baseball-Almanac. (2013). World series history. Retrieved from http://www.baseball-almanac.com/ws/wsmenu.shtml
Graduating Engineer + Computer Careers. (n.d.). Retrieved from http://www.graduatingengineer.com
Microsoft Bookshelf. (n.d.).
Nasdaq. (n.d.). Sectoring by industry groups. Retrieved from http://www.nasdaq.com/markets/barchart-sectors.aspx
U.S. Senate. (n.d.). Retrieved from www.senate.gov
Wikipedia. (n.d.). List of current United States Senators by age. Retrieved from http://en.wikipedia.org/wiki/List_of_current_United_States_Senators_by_age

### Chapter Review

Two population means from independent samples where the population standard deviations are not known
1. Random variable: 
  = the difference of the sampling means
2. Distribution: Student’s t-distribution with degrees of freedom (variances not pooled)

### Formula Review

Standard error: SE = 
    
    
Test statistic (t-score): t = 
 

Degrees of freedom:

 
where:
s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

 and 
 are the sample means.
Cohen’s d is the measure of effect size:

 

where 
 
Use the following information to answer the next 15 exercises. Indicate if the hypothesis test is for
1. independent group means, population standard deviations, and/or variances known,
2. independent group means, population standard deviations, and/or variances unknown,
3. matched or paired samples,
4. single mean,
5. two proportions, or
6. single proportion.
Use the following information to answer the next three exercises: A study is done to determine which of two soft drinks has more sugar. There are 13 cans of Beverage A in a sample and six cans of Beverage B. The mean amount of sugar in Beverage A is 36 grams with a standard deviation of 0.6 grams. The mean amount of sugar in Beverage B is 38 grams with a standard deviation of 0.8 grams. The researchers believe that Beverage B has more sugar than Beverage A, on average. Both populations have normal distributions.
Use the following information to answer the next 12 exercises. The U.S. Centers for Disease Control reports that the mean life expectancy was 47.6 years for whites born in 1900 and 33.0 years for nonwhites. Suppose that you randomly survey death records for people born in 1900 in a certain county. Of the 124 whites, the mean life span was 45.3 years with a standard deviation of 12.7 years. Of the 82 nonwhites, the mean life span was 34.1 years with a standard deviation of 15.6 years. Conduct a hypothesis test to see if the mean life spans in the county were the same for whites and nonwhites.

### Homework

DIRECTIONS: For each of the word problems, use a solution sheet to do the hypothesis test. The solution sheet is found in 
Use the information from 
Use the following information to answer the next two exercises. The Eastern and Western Major League Soccer conferences have a new Reserve Division that allows new players to develop their skills. Data for a randomly picked date showed the following annual goals.
Conduct a hypothesis test to answer the next two exercises.