A soft drink manufacturer wishes to know how many soft drinks teenagers drink each week. They want to construct a 90% confidence interval with an error of no more than 0.07. A consultant has informed them that a previous study found the mean to be 7.4 soft drinks per week and found the variance to be 0.81. What is the minimum sample size required to create the specified confidence interval? Round your answer up to the next integer.

Answer:

Given Data:

A confidence interval $(CI)=0.90$

A margin of error $(E)=0.07$

A population mean $(\mu )=7.4$

A population variance $({\sigma }^2)=0.81$

A minimum sample size for the given data is $=?$

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Solution:

The population standard deviation:

$\sigma =\sqrt{0.81}=0.90$

For a $90$ confidence interval,
The level of significance:

$\alpha =1–CI=1–0.90=0.10$

The critical value:

$Z_c=Z_{\alpha /2}$

$=Z_{0.10/2}$

$=Z_{0.05}$

$\approx 1.645$

Use the formula for margin of error and calculate the sample size:

$E=Z_c\times \frac{\sigma }{\sqrt{n}}$

$0.07=1.645\times \frac{0.90}{\sqrt{n}}$

$n=448$