Question

Independent random samples of

*n*_{1} = 170

and

*n*_{2} = 170

observations were randomly selected from binomial populations 1 and 2, respectively. Sample 1 had 96 successes, and sample 2 had 103 successes.

You wish to perform a hypothesis test to determine if there is a difference in the sample proportions

*p*_{1}

and

*p*_{2}.

(a)

State the null and alternative hypotheses.

*H*_{0}:
(*p*_{1} − *p*_{2})
< 0 versus *H*_{a}:
(*p*_{1} − *p*_{2})
> 0

*H*_{0}:
(*p*_{1} − *p*_{2})
= 0 versus *H*_{a}:
(*p*_{1} − *p*_{2})
≠ 0

*H*_{0}:
(*p*_{1} − *p*_{2})
≠ 0 versus *H*_{a}:
(*p*_{1} − *p*_{2})
= 0

*H*_{0}:
(*p*_{1} − *p*_{2})
= 0 versus *H*_{a}:
(*p*_{1} − *p*_{2})
> 0

*H*_{0}:
(*p*_{1} − *p*_{2})
= 0 versus *H*_{a}:
(*p*_{1} − *p*_{2})
< 0

(b)

Find the test statistic and rejection region, using the
*α* = 0.10 level of significance. (Round your answers to two
decimal places. If the test is one-tailed, enter NONE for the
unused region.)

test statistic*z*=rejection
region*z*>*z*<

(c)

State your conclusion.

*H*_{0} is rejected. There is insufficient
evidence to indicate that *p*_{1} is different from
*p*_{2}.*H*_{0} is rejected. There is
sufficient evidence to indicate that *p*_{1} is
different from
*p*_{2}. *H*_{0}
is not rejected. There is sufficient evidence to indicate that
*p*_{1} is different from
*p*_{2}.*H*_{0} is not rejected.
There is insufficient evidence to indicate that
*p*_{1} is different from
*p*_{2}.

Independent random samples of

*n*_{1} = 170

and

*n*_{2} = 170

observations were randomly selected from binomial populations 1 and 2, respectively. Sample 1 had 96 successes, and sample 2 had 103 successes.

You wish to perform a hypothesis test to determine if there is a difference in the sample proportions

*p*_{1}

and

*p*_{2}.

(a)

State the null and alternative hypotheses.

*H*_{0}:
(*p*_{1} − *p*_{2})
< 0 versus *H*_{a}:
(*p*_{1} − *p*_{2})
> 0

*H*_{0}:
(*p*_{1} − *p*_{2})
= 0 versus *H*_{a}:
(*p*_{1} − *p*_{2})
≠ 0

*H*_{0}:
(*p*_{1} − *p*_{2})
≠ 0 versus *H*_{a}:
(*p*_{1} − *p*_{2})
= 0

*H*_{0}:
(*p*_{1} − *p*_{2})
= 0 versus *H*_{a}:
(*p*_{1} − *p*_{2})
> 0

*H*_{0}:
(*p*_{1} − *p*_{2})
= 0 versus *H*_{a}:
(*p*_{1} − *p*_{2})
< 0

(b)

Find the test statistic and rejection region, using the
*α* = 0.10 level of significance. (Round your answers to two
decimal places. If the test is one-tailed, enter NONE for the
unused region.)

test statistic*z*=rejection
region*z*>*z*<

(c)

State your conclusion.

*H*_{0} is rejected. There is insufficient
evidence to indicate that *p*_{1} is different from
*p*_{2}.*H*_{0} is rejected. There is
sufficient evidence to indicate that *p*_{1} is
different from
*p*_{2}. *H*_{0}
is not rejected. There is sufficient evidence to indicate that
*p*_{1} is different from
*p*_{2}.*H*_{0} is not rejected.
There is insufficient evidence to indicate that
*p*_{1} is different from
*p*_{2}.

Answer #1

a)

H0: (p1 − p2) = 0 versus Ha: (p1 − p2) ≠ 0

b)

p1cap = X1/N1 = 96/170 = 0.5647

p1cap = X2/N2 = 103/170 = 0.6059

pcap = (X1 + X2)/(N1 + N2) = (96+103)/(170+170) = 0.5853

Rejection Region

This is two tailed test, for α = 0.05

Critical value of z are -1.96 and 1.96.

Hence reject H0 if z < -1.96 or z > 1.96

Test statistic

z = (p1cap - p2cap)/sqrt(pcap * (1-pcap) * (1/N1 + 1/N2))

z = (0.5647-0.6059)/sqrt(0.5853*(1-0.5853)*(1/170 + 1/170))

z = -0.77

c)

H0 is not rejected.There is insufficient evidence to indicate that p1 is different from p2.

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