According to an article in Newsweek , the natural ratio of girls to boys is 100:

Question

According to an article in Newsweek, the natural ratio of girls to boys is 100:105. In China, the birth ratio is 100:114 (46.7% girls). Suppose you don't believe the reported figures of the percent of girls born in China. You conduct a study. In this study, you count the number of girls and boys born in 150 randomly chosen recent births. There are 61 girls and 89 boys born of the 150. Based on your study, do you believe that the percent of girls born in China is 46.7? Conduct a hypothesis test at the 5% level.
Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

a) What is the test statistic? (Round your answer to two decimal places.)

b) What is the p-value? (Round your answer to four decimal places.)

c) Construct a 95% confidence interval for the true proportion. Sketch the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval. (Round your answers to four decimal places.)

Explanation / Answer

PART A.
Given that,
possibile chances (x)=61
sample size(n)=150
success rate ( p )= x/n = 0.4067
success probability,( po )=0.467
failure probability,( qo) = 0.533
null, Ho:p=0.467  
alternate, H1: p!=0.467
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 proportion = p-po/sqrt(poqo/n)
zo=0.40667-0.467/(sqrt(0.248911)/150)
zo =-1.4811
| zo | =1.4811
critical value
the value of |z | at los 0.05% is 1.96
we got |zo| =1.481 & | z | =1.96
make decision
hence value of |zo | < | z | and here we do not reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != -1.48109 ) = 0.13858
hence value of p0.05 < 0.1386,here we do not reject Ho
ANSWERS
---------------
null, Ho:p=0.467
alternate, H1: p!=0.467
test statistic: -1.4811
critical value: -1.96 , 1.96
decision: do not reject Ho

PART B.
p-value: 0.13858

hence that the percent of girls born in China is not 46.7.

PART C.
given that,
possibile chances (x)=61
sample size(n)=150
success rate ( p )= x/n = 0.4067
CI = confidence interval
confidence interval = [ 0.4067 ± 1.96 * Sqrt ( (0.4067*0.5933) /150) ) ]
= [0.4067 - 1.96 * Sqrt ( (0.4067*0.5933) /150) , 0.4067 + 1.96 * Sqrt ( (0.4067*0.5933) /150) ]
= [0.3281 , 0.4853]
-----------------------------------------------------------------------------------------------
interpretations:
1. We are 95% sure that the interval [ 0.3281 , 0.4853] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the true population proportion

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