Complete question is;
A study by Consumer Reports showed that 64% of supermarket shoppers believe supermarket
brands to be as good as national name brands. To investigate whether this result applies
to its own product, the manufacturer of a national name-brand ketchup asked a sample
of shoppers whether they believed that supermarket ketchup was as good as the national
brand ketchup.
a. Formulate the hypotheses that could be used to determine whether the percentage of
supermarket shoppers who believe that the supermarket ketchup was as good as the
national brand ketchup differed from 64%.
b. If a sample of 100 shoppers showed 52 stating that the supermarket brand was as good
as the national brand, what is the p-value?
c. At α = 0.05, what is your conclusion?
d. Should the national brand ketchup manufacturer be pleased with this conclusion?
Answer:
A) H0; p =0.64. And HA; p≠0.64
B) p-value = 0.01242
C) Null hypothesis should be rejected
D) No they should not
Step-by-step explanation:
A) The hypotheses are;
Null hypothesis H0; p =0.64.
Alternative hypothesis HA; p≠0.64
B) we are given;
n = 100
x = 52
α = 0.05
The sample proportion is given by; p' = x/n = 52/100 = 0.52
Now let's check np and n(1 - p)
np = 100 x 0.64 = 64
n(1 - p) = 100(1 - 0.64) = 36
They are both greater than 5. So a two sided test
So let's find the value of the test statistic ;
z = [p' - p]/√(p(1 - p)/n)
z = [0.52 - 0.63]/√(0.64(1 - 0.64)/100)
z = - 2.5
Since 2 sided test, thus p value will be;
p = 2P(Z ≤ -2.5)
So from the table i attached,
P(Z ≤ -2.5) = 0.00621
Thus, p = 2 x 0.00621 = 0.01242
C) at α = 0.05, our p value which is 0.01242 is less than α. Thus we reject the null hypothesis.
D) There is evidence to suggest that the percentage of supermarket shoppers who believe that the supermarket Ketchup was as good as the national brand Ketchup differs from64%. Thus, the national brand Ketchup manufacturer should not be pleased with this conclusion, because the results cannot be applied to its own product.
