# Hypothesis Testing with Two Samples

## Comparing Two Independent Population Proportions

When conducting a hypothesis test that compares two independent population proportions, the following characteristics should be present:
1. The two independent samples are simple random samples that are independent.
2.  The number of successes is at least five, and the number of failures is at least five, for each of the samples.
3. Growing literature states that the population must be at least 10 or 20 times the size of the sample. This keeps each population from being over-sampled and causing incorrect results.
Comparing two proportions, like comparing two means, is common. If two estimated proportions are different, it may be due to a difference in the populations or it may be due to chance. A hypothesis test can help determine if a difference in the estimated proportions reflects a difference in the population proportions.
The difference of two proportions follows an approximate normal distribution. Generally, the null hypothesis states that the two proportions are the same. That is, H: p = p. To conduct the test, we use a pooled proportion, p. 
The pooled proportion is calculated as follows:
The distribution for the differences is
The test statistic (z-score) is

### References

American Cancer Society. (n.d.). Retrieved from http://www.cancer.org/index
Centers for Disease Control and Prevention. (n.d.). West Nile virus. Retrieved from http://www.cdc.gov/ncidod/dvbid/westnile/index.htm
Chancellors Office, California Community Colleges. (1994, Nov.).
Educational Resources. (n.d.).
Gallup. (2013). State of the states. Retrieved from http://www.gallup.com/poll/125066/State-States.aspx?ref=interactive
Hilton Hotels. (n.d.). Retrieved from http://www.hilton.com
Hyatt Hotels. Retrieved from http://hyatt.com
San Jose Museum of Art. (n.d.). Whitney exhibit (on loan).
U.S. Department of Health and Human Services. (n.d). Statistics. Retrieved from https://www.hhs.gov/

### Chapter Review

Test of two population proportions from independent samples

  


### Formula Review

Pooled proportion: p = 
    
Distribution for the differences:

 
where the null hypothesis is H: p = p or H: p – p = 0
Test statistic (z-score): 
 
where the null hypothesis is H: p = p or H: p − p = 0
and where
p′
 and p′
 are the sample proportions, p and p are the population proportions,
P is the pooled proportion, and n and n are the sample sizes.
Use the following information for the next five exercises. Two types of phone operating system are being tested to determine if there is a difference in the proportions of system failures (crashes). Fifteen out of a random sample of 150 phones with OS1 had system failures within the first eight hours of operation. Nine out of another random sample of 150 phones with OS2 had system failures within the first eight hours of operation. OS2 is believed to be more stable (have fewer crashes) than OS1.
Use the following information to answer the next 12 exercises. In the recent U.S. Census, 3 percent of the U.S. population reported being of two or more races. However, the percent varies tremendously from state to state. Suppose that two random surveys are conducted. In the first random survey, out of 1,000 North Dakotans, only 9 people reported being of two or more races. In the second random survey, out of 500 Nevadans, 17 people reported being of two or more races. Conduct a hypothesis test to determine if the population percents are the same for the two states or if the percent for Nevada is statistically higher than for North Dakota.


### 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 following information to answer the next three exercises. Neuroinvasive West Nile virus is a severe disease that affects a person’s nervous system. It is spread by the Culex species of mosquito. In the United States in 2010, there were 629 reported  cases of neuroinvasive West Nile virus out of a total of 1,021 reported cases, and there were 486 neuroinvasive reported cases out of a total of 712 cases reported in 2011.  Is the 2011 proportion of neuroinvasive West Nile virus cases more than the 2010 proportion of neuroinvasive West Nile virus cases? Using a 1 percent level of significance, conduct an appropriate hypothesis test.


  