Respuesta :
We need to find out how many adults must the brand manager survey in order to be 90% confident that his estimate is within five percentage points of the true population percentage.
From the given data we know that our confidence level is 90%. From Standard Normal Table we know that the critical level at 90% confidence level is 1.645. In other words, [tex] Z_{Critical}=1.645 [/tex].
We also know that E=5% or E=0.05
Also, since, [tex] \hat{p} [/tex] is not given, we will assume that [tex] \hat{p} [/tex]=0.5. This is because, the formula that we use will have [tex] \hat{p}(1-\hat{p}) [/tex] in the expression and that will be maximum only when [tex] \hat{p} [/tex]=0.5. (For any other value of [tex] \hat{p} [/tex], we will get a value less than 0.25. For example if, [tex] \hat{p} [/tex] is 0.4, then [tex] 1-\hat{p}=0.6 [/tex] and thus, [tex] \hat{p}(1-\hat{p})=0.24 [/tex].).
We will now use the formula
[tex] n=(\frac{Z_{Critical}}{E})^2\hat{p}(1-\hat{p}) [/tex]
We will now substitute all the data that we have and we will get
[tex] n=(\frac{1.645}{0.05})^2\times0.5(1-0.5) [/tex]
[tex] n=(32.9)^2\times0.25 [/tex]
[tex] n=270.6025 [/tex]
which can approximated to n=271.
So, the brand manager needs a sample size of 271
The brand manager needs a sample size of 271
Step-by-step explanation:
Given: The brand manager for a brand toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the percentage of adults Wii have heard of the brand.
As per question:
We have to find out how many adults must the brand manager survey in order to be [tex]90\%[/tex] confident that his estimate is within [tex]5\%[/tex] points of the true population percentage.
As we know that [tex]E=5\%\;\rm{or}\;E=0.05[/tex]
Now, the value of sample size is calculated as [tex]n=(\frac{(Z_{critical})}{E})^2\hat{p}(1-\hat{p})[/tex]
[tex]n=\frac{1.645}{0.05}\times0.5(1-0.5)\\n=(32.9)^2\times0.25\\n=270.6025[/tex]
We can approximate the value of [tex]n[/tex] to [tex]271[/tex].
Therefore, the brand manager needs a sample size of [tex]271[/tex].
Learn more about sample size here:
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