The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the percentage of adults who have heard of the brand. How many adults must he survey in order to be 95% confident that his estimate is within seven percentage points of the true population percentage? Complete parts (a) through (c) below.
a) Assume that nothing is known about the percentage of adults who have heard of the brand,
n= ____________ (Round up to the nearest integer.)
b) Assume that a recent survey suggests that about 81% of adults have heard of the brand.
n= ____________ (Round up to the nearest integer.) Please show work for this one because they change the percentage a lot, so I may have to work this part out myself. Thank you
c) Given that the required sample size is relatively small, cculd he simply survey the adults at the nearest college?
a. No, a sample of students at the nearest college is a cluster sample, not a simple random sample, so it is very possible that the results would not be representative of the population of adults.
b. Yes, a sample of students at the nearest college is a simple random sample, so the resuls should be representative of the population of adults.
c. No, a sample of students at the nearest college is a stratified sample, not a simple random sample, so it is very possible that the results would not be representaive of the population of adults.
d. No, a sample of students at the nearest college is a convenience sample, not a simple randmom sample. so it is very possible that the results would not be representative of the population of adults.

Answer:

Part A;  n= 196

Part B:  n = 120

Part C:

d. No, a sample of students at the nearest college is a convenience sample, not a simple random sample. so it is very possible that the results would not be representative of the population of adults.

Step-by-step explanation:

Part A;

The sample size n can be calculated by using the formula

p^ ± z∝/2* √pq/n

this implies that

e= z∝/2* √pq/n

where e= ║p-p^║= 0.07

p^= 0.5

q^= 0.5

Putting the values

e= z∝/2* √pq/n

0.07= ± 1.96 √0.5*0.5/n

Squaring both sides

0.0049 = 3.8146*0.25/n

n= 196

Part B:

Given p^= 0.81  and q^= 1-p^= 1-0.81= 0.19

e= 0.07

Using the formula

e= z∝/2* √pq/n

Putting the values

0.07= ± 1.96 √0.81*0.19/n

Squaring both sides

0.0049 = 3.8146*0.1539/n

n= 119.809= 120

Part C:

The sample selected is not a random sample because all the adults who have heard about the brand do not have equal chances of being chosen.

It is convenience sample that is it is chosen at ease from a nearby college.

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