Answer:
a) the test statistic z = 1.891
the null hypothesis accepted at 95% level of significance
b) the critical values of 95% level of significance is zα =1.96
c) 95% of confidence intervals are (0.523 ,0.596)
Step-by-step explanation:
A survey of 700 adults from a certain region
Given sample sizes ![n_{1} = 400 and n_{2} = 300](https://tex.z-dn.net/?f=n_%7B1%7D%20%3D%20400%20and%20n_%7B2%7D%20%3D%20300)
Proportion of mean ![p_{1} = \frac{236}{400} = 0.59 and p_{2} = \frac{156}{300} = 0.52](https://tex.z-dn.net/?f=p_%7B1%7D%20%3D%20%5Cfrac%7B236%7D%7B400%7D%20%3D%200.59%20and%20p_%7B2%7D%20%3D%20%5Cfrac%7B156%7D%7B300%7D%20%3D%200.52)
<u>Null hypothesis H0</u> : assume that there is no significant difference between males and women reported they buy clothing from their mobile device
p1 = p2
<u>Alternative hypothesis H1:</u>- p1 ≠ p2
a) The test statistic is
![Z = \frac{p_{1} -p_{2} }{\sqrt{pq(\frac{1}{n_{1} }+\frac{1}{n_{2} )} } }](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7Bp_%7B1%7D%20-p_%7B2%7D%20%7D%7B%5Csqrt%7Bpq%28%5Cfrac%7B1%7D%7Bn_%7B1%7D%20%7D%2B%5Cfrac%7B1%7D%7Bn_%7B2%7D%20%29%7D%20%20%7D%20%7D)
where
on calculation we get p = 0.56
now q =1-p = 1-0.56=0.44
![Z = \frac{p_{1} -p_{2} }{\sqrt{pq(\frac{1}{n_{1} }+\frac{1}{n_{2} )} } }\\ =\frac{0.56-0.52}{\sqrt{0.56X0.44}(\frac{1}{400}+\frac{1}{300} }](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7Bp_%7B1%7D%20-p_%7B2%7D%20%7D%7B%5Csqrt%7Bpq%28%5Cfrac%7B1%7D%7Bn_%7B1%7D%20%7D%2B%5Cfrac%7B1%7D%7Bn_%7B2%7D%20%29%7D%20%20%7D%20%7D%5C%5C%20%20%20%3D%5Cfrac%7B0.56-0.52%7D%7B%5Csqrt%7B0.56X0.44%7D%28%5Cfrac%7B1%7D%7B400%7D%2B%5Cfrac%7B1%7D%7B300%7D%20%20%20%7D)
after calculation we get z = 1.891
b) The critical value at 95% confidence interval zα = 1.96 (from z-table)
The calculated z- value < the tabulated value
therefore the null hypothesis accepted
<u>conclusion</u>:-
assume that there is no significant difference between males and women reported they buy clothing from their mobile device
p1 = p2
c) <u>95% confidence intervals</u>
The confidence intervals are P± 1.96(√PQ/n)
we know that = ![p = \frac{n_{1}p_{1} +n_{2}p_{2} }{n_{1}+n_{2}}= \frac{400X0.59+300X0.52}{700}](https://tex.z-dn.net/?f=p%20%3D%20%5Cfrac%7Bn_%7B1%7Dp_%7B1%7D%20%2Bn_%7B2%7Dp_%7B2%7D%20%7D%7Bn_%7B1%7D%2Bn_%7B2%7D%7D%3D%20%5Cfrac%7B400X0.59%2B300X0.52%7D%7B700%7D)
after calculation we get P = 0.56 and Q =1-P =0.44
Confidence intervals are ( P- 1.96(√PQ/n), P+ 1.96(√PQ/n))
now substitute values , we get
( 0.56- 1.96(√0.56X0.44/700), 0.56+ 1.96(0.56X0.44/700))
on simplification we get (0.523 ,0.596)
Therefore the population proportion (0.56) lies in between the 95% of <u>confidence intervals (0.523 ,0.596)</u>
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