A study conducted by the non-profit Safe Kids campaign in 2004 found that 33% of
ID: 2925203 • Letter: A
Question
A study conducted by the non-profit Safe Kids campaign in 2004 found that 33% of kids riding a bicycle on a residential street wore a safety helmet. Suppose that you watch a sample of 30 kids riding a bike on a residential street in your community. Let the random variable H represent the number of those 30 kids who are wearing a helmet.
(a) Would H follow a binomial distribution? Explain.
(b) If the rate of helmet wearing is the same in your community as in the Safe Kids study, how many kids would you expect to be wearing a helmet?
(c) Explain how you might use a six-sided die to simulate the distribution of H.
(d) Use R code to simulate 10,000 sets of dice rolls and compute the probability of at least 12 kids wearing helments.
(d) Suppose that 15 of the 30 kids in your sample are wearing a helmet. Is that count large enough to convince you, at the 10% significance level, that the helmet-wearing rate in your community is higher than 33%? Address this question with all of the steps of a hypothesis test of significance.
Explanation / Answer
a.
A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties:
1.The experiment consists of n repeated trials.
2.Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure.
3.The probability of success, denoted by P, is the same on every trial.
4.The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.
H follows BINOMIAL DISTRIBUTION
b.
pmf of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
where
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial
I.
Expect to be wearing a helmet= mean = np
where
n = total number of repetitions experiment is excueted
p = success probability
mean = 30 * 0.33
= 9.9
d.
Given that,
possibile chances (x)=15
sample size(n)=30
success rate ( p )= x/n = 0.5
success probability,( po )=0.33
failure probability,( qo) = 0.67
null, Ho:p=0.33
alternate, H1: p>0.33
level of significance, = 0.1
from standard normal table,right tailed z /2 =1.28
since our test is right-tailed
reject Ho, if zo > 1.28
we use test statistic z proportion = p-po/sqrt(poqo/n)
zo=0.5-0.33/(sqrt(0.2211)/30)
zo =1.9802
| zo | =1.9802
critical value
the value of |z | at los 0.1% is 1.28
we got |zo| =1.98 & | z | =1.28
make decision
hence value of | zo | > | z | and here we reject Ho
p-value: right tail - Ha : ( p > 1.98023 ) = 0.02384
hence value of p0.1 > 0.02384,here we reject Ho
ANSWERS
---------------
null, Ho:p=0.33
alternate, H1: p>0.33
test statistic: 1.9802
critical value: 1.28
decision: reject Ho
p-value: 0.02384
enough evidence to support that the helmet-wearing rate in your community is higher than 33%
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