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Cerrena [4.2K]
3 years ago
10

Ou are trying to estimate the average amount a family spends on food during a year. In the past, the standard deviation of the a

mount a family has spent on food during a year has been σ = $1200. If you want to be 99% sure that you have estimated average family food expenditures within $60, how many families do you need to survey? Place your answer, a whole number, do not use any decimals, in the blank___.
Mathematics
1 answer:
MA_775_DIABLO [31]3 years ago
5 0

Answer:

The answer is "2653".

Step-by-step explanation:

Sample \ Size = 99\% \ Cl\\\\ E =60  \\\\\sigma= 1200\\\\

Minimum sample size for Cl level = 99\% and Desired Margin of Error, E = 60 is:  

\to P(|\bar{x}-\mu| < E) \geq 1-\alpha \\\\\to P(\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}

Margin \  Error = \frac{\text{length of the CI}}{2}\\\\\sigma=1200\\\\\alpha=1 - Confidence =1-0.99=0.01\\\\Z_{\frac{\alpha}{2}}=z_{0.005}=2.58\\\\n\geq (\frac{z_{0.005} \times \sigma }{E})^2 \\\\= (\frac{2.58 \times 1200 }{60})^2\\\\=2662.560\\

The minimum n has to be integer, we take the ceiling Of above number and get n = 2663

The exact z-value.

n \geq (\frac{2.5758293035489 \times 1200}{60})^2 \\\\= 2653.95864

using critical value of 2,575, which gives 2652.25  

\to n =2653

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