<< Chapter < Page Chapter >> Page >

Try it

A concerned group of citizens wanted to know if the proportion of forcible rapes in Texas was different in 2011 than in 2010. Their research showed that of the 113,231 violent crimes in Texas in 2010, 7,622 of them were forcible rapes. In 2011, 7,439 of the 104,873 violent crimes were in the forcible rape category. Test at a 5% significance level. Answer the following questions:

a. Is this a test of two means or two proportions?

a. two proportions

Got questions? Get instant answers now!

b. Which distribution do you use to perform the test?

b. normal for two proportions

Got questions? Get instant answers now!

c. What is the random variable?

c. Subscripts: 1 = 2010, 2 = 2011
P 2 - P 2

Got questions? Get instant answers now!

d. What are the null and alternative hypothesis? Write the null and alternative hypothesis in symbols.

d. Subscripts: 1 = 2010, 2 = 2011
H 0 : p 1 = p 2 H 0 : p 1 p 2 = 0
H a : p 1 p 2 H a : p 1 p 2 ≠ 0

Got questions? Get instant answers now!

e. Is this test right-, left-, or two-tailed?

e. two-tailed

Got questions? Get instant answers now!

f. What is the p -value?

f. p -value = 0.00086

This is a normal distribution curve with mean equal to zero. Both the right and left tails of the curve are shaded. Each tail represents 1/2(p-value) = 0.0004.
Got questions? Get instant answers now!

g. Do you reject or not reject the null hypothesis?

g. Reject the H 0 .

Got questions? Get instant answers now!

h. At the ___ level of significance, from the sample data, there ______ (is/is not) sufficient evidence to conclude that ____________.

h. At the 5% significance level, from the sample data, there is sufficient evidence to conclude that there is a difference between the proportion of forcible rapes in 2011 and 2010.

Got questions? Get instant answers now!

References

Data from Educational Resources , December catalog.

Data from Hilton Hotels. Available online at http://www.hilton.com (accessed June 17, 2013).

Data from Hyatt Hotels. Available online at http://hyatt.com (accessed June 17, 2013).

Data from Statistics, United States Department of Health and Human Services.

Data from Whitney Exhibit on loan to San Jose Museum of Art.

Data from the American Cancer Society. Available online at http://www.cancer.org/index (accessed June 17, 2013).

Data from the Chancellor’s Office, California Community Colleges, November 1994.

“State of the States.” Gallup, 2013. Available online at http://www.gallup.com/poll/125066/State-States.aspx?ref=interactive (accessed June 17, 2013).

“West Nile Virus.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/ncidod/dvbid/westnile/index.htm (accessed June 17, 2013).

Chapter review

Test of two population proportions from independent samples.

  • Random variable: p ^ A p ^ B = difference between the two estimated proportions
  • Distribution: normal distribution

Formula review

Pooled Proportion: p c = x F   +   x M n F   +   n M

Distribution for the differences:
p A p B N [ 0 , p c ( 1 p c ) ( 1 n A + 1 n B ) ]

where the null hypothesis is H 0 : p A = p B  or  H 0 : p A p B = 0.

Test Statistic ( z -score): z = ( p A p B ) p c ( 1 p c ) ( 1 n A + 1 n B )

where the null hypothesis is H 0 : p A = p B  or  H 0 : p A p B = 0.

where

p′ A and p′ B are the sample proportions, p A and p B are the population proportions,

P c is the pooled proportion, and n A and n B 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 OS 1 had system failures within the first eight hours of operation. Nine out of another random sample of 150 phones with OS 2 had system failures within the first eight hours of operation. OS 2 is believed to be more stable (have fewer crashes) than OS 1 .

Is this a test of means or proportions?

Got questions? Get instant answers now!

What is the random variable?

P OS1 P OS2 = difference in the proportions of phones that had system failures within the first eight hours of operation with OS 1 and OS 2 .

Got questions? Get instant answers now!

State the null and alternative hypotheses.

Got questions? Get instant answers now!

What is the p -value?

0.1018

Got questions? Get instant answers now!

What can you conclude about the two operating systems?

Got questions? Get instant answers now!


Use the following information to answer the next twelve exercises. In the recent Census, three 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 nine 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.

Is this a test of means or proportions?

proportions

Got questions? Get instant answers now!

State the null and alternative hypotheses.

  1. H 0 : _________
  2. H a : _________
Got questions? Get instant answers now!

Is this a right-tailed, left-tailed, or two-tailed test? How do you know?

right-tailed

Got questions? Get instant answers now!

What is the random variable of interest for this test?

Got questions? Get instant answers now!

In words, define the random variable for this test.

The random variable is the difference in proportions (percents) of the populations that are of two or more races in Nevada and North Dakota.

Got questions? Get instant answers now!

Which distribution (normal or Student's t ) would you use for this hypothesis test?

Got questions? Get instant answers now!

Explain why you chose the distribution you did for the Exercise 10.56 .

Our sample sizes are much greater than five each, so we use the normal for two proportions distribution for this hypothesis test.

Got questions? Get instant answers now!

Calculate the test statistic.

Got questions? Get instant answers now!

Sketch a graph of the situation. Mark the hypothesized difference and the sample difference. Shade the area corresponding to the p -value.

This is a horizontal axis with arrows at each end. The axis is labeled p'N - p'ND

Check student’s solution.

Got questions? Get instant answers now!

At a pre-conceived α = 0.05, what is your:

  1. Decision:
  2. Reason for the decision:
  3. Conclusion (write out in a complete sentence):

  1. Reject the null hypothesis.
  2. p -value<alpha
  3. At the 5% significance level, there is sufficient evidence to conclude that the proportion (percent) of the population that is of two or more races in Nevada is statistically higher than that in North Dakota.
Got questions? Get instant answers now!

Does it appear that the proportion of Nevadans who are two or more races is higher than the proportion of North Dakotans? Why or why not?

Got questions? Get instant answers now!

Questions & Answers

how sampling error reduced
Eman Reply
what is the test hpothesis of 100 students from five schools using instructional materials
ngozi Reply
what is the test hypothesis of 100 students using instructional materials and ten teachers from five schools
ngozi
what is test hypothesis of 10 teachers using instructional materials to teach 100 students from five schools
ngozi
1. Six (6) “Yes” and “No” questions were modelled for fifty randomly selected student in a class. a) What is the probability that a student will get at least three (3) questions correct? b) What is the probability that a student will get all the questions correct? c) How may students are expected to get three (3) or more question correct
Joshua Reply
observation is the art of facts. justify the statment with proper arguments
Sheikh
how to calculate prevalence
Ayesha Reply
how do I differentiate between descriptive and inferential statistics?
betuel Reply
observation is the art of facts. justify the statment with proper arguments
Sheikh
The “average increase” for all NASDAQ stocks is the
Jacklyn Reply
Find the probability that an Emotional Health Index Score is between 80.5 and 82?
Mounika Reply
how to solve probability questions
Eke Reply
Calaulate the radins of gyration (k for the shaded aren about the vas. (Dimemions in mm) +y- 50
Material
chijioke
How do you get log of normal population
Shan Reply
plz give me answer of this question show that mean of population or sampling distribution are equally
Mari
yes
Hazrat
what is the probability of getting no head face up in three tosses of a fair coin?
Epara Reply
In how many ways can probability be assigned to an event of interest?
Epara
Hey guys can someone help me with combinations and permutations
Lion
1/8
Mounika
hey guys i wish d best
rasheed
wish u the best too
Chisomo
wish u the best too
Chisomo
Phone lines on an airline system are occupied 50% of time assume that 10 calls are placed to the airline. What is the probability at least 1 call the lines are occupied?
Highness Reply
Phone lines on an airline system are occupied 50% of time assume that 10 calls are placed to the airline. What is the probability at least 1 call the lines are occupied?
Highness Reply
Why is the method of selecting the sample even more important than the size of the sample?
Nana Reply
formulaas for gruoped and ungrouped of quartiles
chatered Reply
Why is the method of selecting the sample even more important than the size of the sample?
Nana
una empresa productora que participa en el mercado de lapices tiene la siguente funcion de demanda Qd=a la raiz de 9 -9p si la elasticidad precio de demanda de la empresa es epd=0,5 determinar a) hallar el precio y cantidad
Jany Reply

Get the best Introductory statistics course in your pocket!





Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Introductory statistics' conversation and receive update notifications?

Ask