Question

A fast-food restaurant promotes certain food items by giving a game piece with each item. Advertisements proclaim that “25% of the game pieces are Instant Winners!” To test this claim, a frequent diner collects 20 game pieces and gets only 3 instant winners.

1. Identify the population, the parameter, the sample, and the statistic in this context.

Suppose the advertisements are correct and p = 0.25. The dotplot below shows the distribution of the sample proportion of instant winners in 100 simulated SRSs of size n = 20.

2. Would it be unusual to get a sample proportion of = 3/20 = 0.15 or less in a sample of size n = 20 when p = 0.25? Explain.

3. Based on your answer to part 2, is there convincing evidence that fewer than 25% of all game pieces are instant winners? Explain.

Answer 1

Given that,
possibile chances (x)=3
sample size(n)=20
success rate ( p )= x/n = 0.15
success probability,( po )=0.25
failure probability,( qo) = 0.75
null, Ho:p=0.25
alternate, H1: p<0.25
level of significance, α = 0.05
from standard normal table,left tailed z α/2 =1.645
since our test is left-tailed
reject Ho, if zo < -1.645
we use test statistic z proportion = p-po/sqrt(poqo/n)
zo=0.15-0.25/(sqrt(0.1875)/20)
zo =-1.033
| zo | =1.033
critical value
the value of |z α| at los 0.05% is 1.645
we got |zo| =1.033 & | z α | =1.645
make decision
hence value of |zo | < | z α | and here we do not reject Ho
p-value: left tail - Ha : ( p < -1.0328 ) = 0.15085
hence value of p0.05 < 0.15085,here we do not reject Ho
ANSWERS
---------------
null, Ho:p=0.25
alternate, H1: p<0.25
test statistic: -1.033
critical value: -1.645
decision: do not reject Ho
p-value: 0.15085
we do not have enough evidence to support the claim that fewer than 25% of all game pieces are instant winners.