Solved on Nov 25, 2023

Estimate the mean water usage in gallons per day for a small town, with a maximum error of 0.15 gallons and 80% confidence. Given that the standard deviation is 2.3 gallons and the mean is 17.6 gallons, determine the required sample size.

STEP 1

Assumptions1. The mean household usage of water is17.6 gallons per day. . The standard deviation is.3 gallons.
3. The maximum error is0.15 gallons.
4. The level of confidence is80%.

STEP 2

We will use the formula for the sample size in a confidence interval estimation. The formula isn=(Zα/2σ)2n = \left(\frac{Z_{\alpha/2} \cdot \sigma}{}\right)^2where- nn is the sample size- Zα/2Z_{\alpha/2} is the Z-score corresponding to the desired level of confidence- σ\sigma is the standard deviation- $$ is the maximum error

STEP 3

First, we need to find the Z-score corresponding to the80% level of confidence. The Z-score for an80% level of confidence is1.28 (you can find this value in a standard Z-table or using a statistical calculator).

STEP 4

Now, plug in the given values into the formula to calculate the sample size.
n=(1.282.30.15)2n = \left(\frac{1.28 \cdot2.3}{0.15}\right)^2

STEP 5

Calculate the sample size.
n=(1.282.30.15)2=245.97n = \left(\frac{1.28 \cdot2.3}{0.15}\right)^2 =245.97

STEP 6

Since we can't have a fraction of a sample, we need to round up to the next integer. Therefore, the required sample size is246.
The water works commission needs a sample size of246 to estimate the mean usage of water with a maximum error of0.15 gallons at an80% level of confidence.

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