In 2015, it was found that 64% of Americans owned a smartphone (according to Pew

Question

In 2015, it was found that 64% of Americans owned a smartphone (according to Pew Research Group). Suppose we are going to take a sample of 20 Americans from each of the fifty states.

Find the sampling distribution of the statistic of interest. (2 points)

Based on this sampling distribution from (a), what is the probability that we find that between 625 and 685 of the Americans sampled own a smartphone? (4 points)

Now in 2016, we found from our sample that 678 Americans owned a smartphone.   Construct a 96% confidence interval to estimate the percentage of Americans who own a smartphone in 2016. Be sure to show all steps.    (5 points)

Based on your interval from (c), could we conclude that more Americans own a smartphone in 2016 than they did in 2015? Explain. (1 point)

Does gender influence owning a smart phone? From our sample above, we also recorded if the respondent was male or female. See dataset: smartphone.xls. Conduct the appropriate test to see if gender influences smartphone ownership in 2016. Be sure to show all steps and use a 5% significance level. (Do test statistic by hand, SC for p-value only!!) (and include appropriate output) (6 points)

Now using the information from part (e), create two pie-charts (one for each gender) to compare those that own a smartphone to those who do not own a smartphone. Does there appear to be a visual difference between these two graphs? Explain (noting how your decision in part (e) above is related).   Include output here. (3 points)

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Explanation / Answer

PART A.
NORMAL APPROXIMATION TO BINOMIAL DISTRIBUTION
the PDF of normal distribution is = 1/ * 2 * e ^ -(x-u)^2/ 2^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
mean ( np ) = 1000 * 0.64 = 640
standard deviation ( npq )= 1000*0.64*0.36 = 15.1789
equation of the normal curve is ( Z )= x - u / sd/sqrt(n) ~ N(0,1)
a.
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 625) = (625-640)/15.1789
= -15/15.1789 = -0.9882
= P ( Z <-0.9882) From Standard Normal Table
= 0.16152
P(X < 685) = (685-640)/15.1789
= 45/15.1789 = 2.9646
= P ( Z <2.9646) From Standard Normal Table
= 0.99848
P(625 < X < 685) = 0.99848-0.16152 = 0.837

b.
TRADITIONAL METHOD
given that,
possible chances (x)=678
sample size(n)=1000
success rate ( p )= x/n = 0.678
I.
sample proportion = 0.678
standard error = Sqrt ( (0.678*0.322) /1000) )
= 0.0148
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.04
from standard normal table, two tailed z /2 =2.054
margin of error = 2.054 * 0.0148
= 0.0303
III.
CI = [ p ± margin of error ]
confidence interval = [0.678 ± 0.0303]
= [ 0.6477 , 0.7083]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
possible chances (x)=678
sample size(n)=1000
success rate ( p )= x/n = 0.678
CI = confidence interval
confidence interval = [ 0.678 ± 2.054 * Sqrt ( (0.678*0.322) /1000) ) ]
= [0.678 - 2.054 * Sqrt ( (0.678*0.322) /1000) , 0.678 + 2.054 * Sqrt ( (0.678*0.322) /1000) ]
= [0.6477 , 0.7083]
-----------------------------------------------------------------------------------------------
interpretations:
1. We are 96% sure that the interval [ 0.6477 , 0.7083] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 96% of these intervals will contains the true population proportion

PART C.
Based on your interval from (c), could we conclude that more Americans own a smartphone
in 2016 than they did in 2015. since the CI is achieved is more than [0.648 , 0.708]

PART D.
Given that,
sample one, male =575, n1 =1000, p1= x1/n1=0.575
sample two, female =424, n2 =1000, p2= x2/n2=0.424
null, Ho: p1 = p2
alternate, H1: p1 != p2
level of significance, = 0.05
from standard normal table, two tailed z /2 =1.96
since our test is two-tailed
reject Ho, if zo < -1.96 OR if zo > 1.96
we use test statistic (z) = (p1-p2)/(p^q^(1/n1+1/n2))
zo =(0.575-0.424)/sqrt((0.5*0.501(1/1000+1/1000))
zo =6.753
| zo | =6.753
critical value
the value of |z | at los 0.05% is 1.96
we got |zo| =6.753 & | z | =1.96
make decision
hence value of | zo | > | z | and here we reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != 6.7529 ) = 0
hence value of p0.05 > 0,here we reject Ho
ANSWERS
---------------
null, Ho: p1 = p2
alternate, H1: p1 != p2
test statistic: 6.753
critical value: -1.96 , 1.96
decision: reject Ho
p-value: 0

have evidence that gender influences smartphone ownership in 2016

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