# Single Proportion and Hypothesis Testing of Equality of two Means

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Sub: Statistics                                                                     Topic: Hypothesis Testing

Single Proportion and Hypothesis Testing of Equality of two Means

Question:
1. National figures reveal that the probability of a youth offender on probation committing another
crime is 0.80. (This is called “recidivism.”) After a special government program intended to reduce
recidivism is put in place, a sample of 100 young offenders enrolled in the program reveals that 70
subsequently committed another crime. You are asked to make a judgment about whether the
government funds invested in this program have succeeded in reducing crime.

a. State the null and alternative hypotheses.

b. Compute the value of the test statistic.

c. Taking alpha=.01, conduct a hypothesis test using the p-value method.

2. Independent random samples of 400 observations each were selected from two populations, giving
the following results:

Sample 1                                  Sample 2

x1 = 5,275                               x2 = 5,240

s1=150                                   s2=200

You want to know whether the population means are different in the two populations. Conduct an
hypothesis test using an alpha of .05 and the critical value method. (Be sure to clearly state your null
and alternative hypothesis, the value of your test statistic, the critical value(s) you compare it to, and

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Sub: Statistics                                                                       Topic: Hypothesis Testing

Solution:
1. The information given in the above problem can be represented with the following notation:
Sample size: n = 100 young offenders enrolled
Number of response: X = 70
Level of significance = α = 0.01

Thus, the sample proportion is given by
p=

70
=
100
= 0.7
Thus the value of q = 1 – p
= 1 – 0.7
= 0.3
Hypotheses:
Null hypothesis:
H0: P0 = 0.80
That is, the government funds invested in the program do not succeeded in reducing crime
Alternative hypothesis:
H1: P0 < 0.80           (One tail Left tail test)
That is, the government funds invested in this program have succeeded in reducing crime
Test Statistic:
p    P0
Z=
P0 Q0
n

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Sub: Statistics                                                                     Topic: Hypothesis Testing
Where
Sample proportion = p = 0.70
Population proportion to be tested = P0 = 0.80
Qo= 1-P0 = 0.20
On substituting the respective values, we have
0.70 0.80
Z=
0.80 * 0.20
100
0.10
=         = -2.5
0.04
Thus the value of test statistic is Z = - 2.5
Critical value:
The critical value of Z statistics at α = 0.01 level of significance is -2.326
Critical region:
The critical region (Left-tailed test) of Z at α = 0.01 level of significance is given by
C = {x / Z < - 2.326}
Conclusion (by critical value approach)
Since the value of test statistic (- 2.5 < -2.326) lies in the critical region, we reject the null hypothesis
at 1% level of significance. Hence, we conclude that the government funds invested in the program
have succeeded in reducing crime.
P-value:
P-value = P [Z < -2.5] = 0.00621 (lower tail) by referring standard normal table.

Conclusion:
Since the p-value (0.00621) is less than 0.01, we reject the null hypothesis at 1% level of significance.
Hence, we conclude that the government funds invested in this program have succeeded in

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Sub: Statistics                                                                     Topic: Hypothesis Testing
reducing crime.

2. The information given in the above problem can be represented as follows:

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 views: 55 posted: 3/6/2013 language: English pages: 7
Description: "1. National figures reveal that the probability of a youth offender on probation committing another crime is 0.80. (This is called “recidivism.”) After a special government program intended to reduce recidivism is put in place, a sample of 100 young offenders enrolled in the program reveals that 70 subsequently committed another crime. You are asked to make a judgment about whether the government funds invested in this program have succeeded in reducing crime. a. State the null and alternative hypotheses. b. Compute the value of the test statistic. c. Taking alpha=.01, conduct a hypothesis test using the p-value method. "
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