A survey of Internet users reported that 21% downloaded music onto their compute

Question

A survey of Internet users reported that 21% downloaded music onto their computers. The filing of lawsuits by the recording industry may be a reason why this percent has decreased from the estimate of 33% from a survey taken two years before Suppose we are not exactly sure about the sizes of the samples. Perform the calculations for the significance tests and 95% confidence intervals under each of the following assumptions. (Use previous recent. Round your test statistics to two decimal places and your confidence intervals to four decimal places.) (i) Both sample sizes are 1000. 0.08 95% C X 0.16 (ii) Both sample sizes are 1600. 7,72 95% (iii) The sample size for the survey reporting 33% is 1 and the sample size for the survey reporting 21% is 1600 000 99% CI. Summarize the effects of the sample sizes on the results O We see in (i) and (ii) that smaller samples result in smaller z (weaker evidence) and narrower intervals, while larger samples have the reverse effect. The results of (iii) show that the effect of varying unequal sample sizes is more complicated. O We see in (i) and (ii) that smaller samples result in smaller z (weaker evidence) and wider intervals, while larger samples have the reverse effect. The results of (iii) show that the effect of varying unequal sample sizes is more complicated. We see in (i) and (ii) that smaller samples result in smaller z (stronger evidence) and wider intervals, while larger samples have the reverse effect. The results of (iii) show that the effect of varying unequal sample sizes is more complicated. O We see in (i) and (ii) that smaller samples result in larger 2 (stronger evidence) and narrower intervals, while larger samples have the reverse effect. The results of (iii) show that the effect of varying unequal sample sizes is more complicated. O We see in (i) and (ii) that smaller samples result in larger z weaker evidence and smaller intervals while larger samples have the reverse effect. The results of (iii) show that the effect of varying unequal sample sizes is more complicated.

Explanation / Answer

(i) H0 : p1 = p2

H1 : ;p1 > p2

p1 = 33% and p2 = 21% and n = 1000

The mean of the difference between both proportions = p1 - p2 = 0.33 - 0.21 = 0.12

p1 = 33% and p2 = 21% and n = 1000

p* = (330 + 210) / (1000 + 1000) = 0.27

Where n1 =n2 = 1000

Z = ( 0.33 - 0.21) / sqrt [ 0.27* 0.73/(1/1000 + 1/1000)] = 6.04

confidence interval = (p1p2) ± z/2 [s.e.(p1p2)]

where s.e.(p1p2) = sqrt [ p1 ( 1- p1 ) /n1 + p2 ( 1- p2 ) /n2] = sqrt[ 0.33 * 0.67/1000 + 0.21 * 0.79 /1000]

= 0.01967

confidence inteval = 0.12 +- 1.96 [ 0.01967] =( 0.0815, 0.1585)

(ii) H0 : p1 = p2

H1 : ;p1 >  p2

p1 = 33% and p2 = 21% and n = 1600

The mean of the difference between both proportions = p1 - p2 = 0.33 - 0.21 = 0.12

p1 = 33% and p2 = 21% and n1 = n2 = 1600

p* = (528 + 336) / (1600 + 1600) = 0.27

Where n1 =n2 = 1600

Z = (p1 - p2 )/ sqrt [ p* ( 1-p*) (1/n1 + 1/n2 )] = ( 0.33 - 0.21) / sqrt [ 0.27* 0.73/(1/1600 + 1/1600)] = 7.65

confidence interval =  (p1p2) ± z/2 [s.e.(p1p2)]

where s.e.(p1p2) = sqrt [ p1 ( 1- p1 ) /n1 + p2 ( 1- p2 ) /n2] = sqrt[ 0.33 * 0.67/1600 + 0.21 * 0.79 /1600]

= 0.01555

confidence inteval = 0.12 +- 1.96 [ 0.01555] =( 0.0895, 0.1505)

(c) H0 : p1 = p2

H1 : ;p1 >  p2

p1 = 33% and p2 = 21% and n1 = 1000 and n2 = 1600

The mean of the difference between both proportions = p1 - p2 = 0.33 - 0.21 = 0.12

p1 = 33% and p2 = 21% a

p* = (330 + 336) / (1000 + 1600) = 0.2562

Z = (p1 - p2 )/ sqrt [ p* ( 1-p*) (1/n1 + 1/n2 )] = ( 0.33 - 0.21) / sqrt [ 0.2562* 0.7438/(1/1000 + 1/1600)] = 6.82

confidence interval =  (p1p2) ± z/2 [s.e.(p1p2)]

where s.e.(p1p2) = sqrt [ p1 ( 1- p1 ) /n1 + p2 ( 1- p2 ) /n2] = sqrt[ 0.33 * 0.67/1000 + 0.21 * 0.79 /1600]

= 0.01802

confidence inteval = 0.12 +- 1.96 [ 0.01802] =( 0.0847, 0.1553)

SO out of the 5 options

Here we see that lower sample size create lower z - values that mean weaker evidence and wider intervals. While larger samples have reverse effect. Reuslts of varying sample sizes are complicated.

so option (2) is correct.

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