Question

1)A professor using an open-source introductory statistics book predicts that 60% of the students will purchase a hard copy of the book, 25% will print it out from the web, and 15% will read it online. She has 200 students this semester. Assuming that the professor's predictions were correct, calculate the expected number of students who read the book online.

2)The number of AIDS cases reported for Santa Clara County, California is broken down by race in the table below. Source: "HIV/AIDS Epidemiology Santa Clara County", Santa Clara County Public Health Department, May 2011.

Race	Cases
White	2136
Hispanic	1122
Black	448
Asian/Pacific Islander	227
Total	3933

Directions: Conduct a chi-square test for goodness-of-fit to determine whether or not the occurrence of AIDS cases is consistent with the race distribution of Santa Clara County.

1. Choose the correct null and alternative hypotheses.
   • H0:H0: The distribution of AIDS cases is consistent with the race distribution in Santa Clara County.
     HaHa The distribution of AIDS cases is different from the race distribution in Santa Clara County.
   • H0H0 The distribution of AIDS cases is different from the race distribution in Santa Clara County.
     Ha:Ha: The distribution of AIDS cases is consistent with the race distribution in Santa Clara County.
2. Compute the test statistic.
   
   The population distribution of Santa Clara County by race is provided in the table below. Use these percentages to compute the expected number of cases for each racial group. Round each of the expected counts to 2 decimal places.
   

   Race	Proportion	Expected cases
   White	0.424
   Hispanic	0.259
   Black	0.025
   Asian/Pacific Islander	0.292
   Total	1

   
   Determine the value of the test statistic. Round your answer to 1 decimal place.
   
   χ2=χ2=
   
3. Compute the p-value. Round your answer to 4 decimal places.
   
   p-value =
   
4. Interpret the results of the significance test.
   • The differences between the distribution of AIDS cases and the distribution of the general population in Santa Clara County are not statistically significant. From a practical perspective, the differences are minor.
   • The differences between the distribution of AIDS cases and the distribution of the general population in Santa Clara County is statistically significant. The differences are also important from a practical perspective. For example, the number of blacks with AIDS is 3.6 times more than expected and the number of Asian/Pacific Islanders with AIDS is 4.1 times less than expected.

Answer 1

1)A professor using an open-source introductory statistics book predicts that 60% of the students will purchase a hard copy of the book, 25% will print it out from the web, and 15% will read it online. She has 200 students this semester. Assuming that the professor's predictions were correct, calculate the expected number of students who read the book online.

Whenever we have to calulate an exoected value we always multiply the total frequnecy with the respective probabilities because expectations are round the accurate values with some added probability.

Books	P	Frequency  ( P * 200)
Hard copy	60%	120
Printed	25%	50
Online	15%	30
	100%	200

2)The number of AIDS cases reported for Santa Clara County, California is broken down by race in the table below. Source: "HIV/AIDS Epidemiology Santa Clara County", Santa Clara County Public Health Department, May 2011.

This is the observed data or the frequencies

Race	Cases (Oi)
White	2136
Hispanic	1122
Black	448
Asian/Pacific Islander	227
Total	3933

Conduct a chi-square test for goodness-of-fit to determine whether or not the occurrence of AIDS cases is consistent with the race distribution of Santa Clara County.

Whenever there is a goodness of fit test we always believe that is the null hypothesis the data is consisttent no difference between the expected and the observed values.

1. Choose the correct null and alternative hypotheses.
   • H0: The distribution of AIDS cases is consistent with the race distribution in Santa Clara County.
     Ha The distribution of AIDS cases is different from the race distribution in Santa Clara County.
   • -
2. Compute the test statistic.

Test Stat =

The population distribution of Santa Clara County by race is provided in the table below. Use these percentages to compute the expected number of cases for each racial group. Round each of the expected counts to 2 decimal places. Population is obtained from the old the data so this forms our expected data.

Again the expected frequencies = Proportion * Total freq.

Here the total freq is taken from the total of the observed freq.

Race	Proportion	Expected cases P * 3933 (Ei)	Cases (Oi)	Oi - Ei
White	0.424	1667.592	2136	468.408	219406.1	131.5706
Hispanic	0.259	1018.647	1122	103.353	10681.84	10.4863
Black	0.025	98.325	448	349.675	122272.6	1243.556
Asian/PI	0.292	1148.436	227	-921.436	849044.3	739.3048
Total	1	3933	3933			2124.917

Determine the value of the test statistic. Round your answer to 1 decimal place.

Compute the p-value. Round your answer to 4 decimal places.

p-value = P( > Test Stat ) ................Where n =no. of categories and df = n-1 = 3

= P( > 2124.917)

..............using chi -dsist tables with df = 3

  

Interpret the results of the significance test.

Since p-value < level of significance (any 0.1,0.05, 0.01)

We reject the null hypothesis. That means there is significant difference.

• The differences between the distribution of AIDS cases and the distribution of the general population in Santa Clara County is statistically significant. The differences are also important from a practical perspective. For example, the number of blacks with AIDS is 3.6 times more than expected and the number of Asian/Pacific Islanders with AIDS is 4.1 times less than expected.