For 45,000 households in a particular county, the water use (in thousands of gal
ID: 3252073 • Letter: F
Question
For 45,000 households in a particular county, the water use (in thousands of gallons) over the course of a year was recorded. The mean water use for the households in the country was found to be 162 and the standard deviation was 20. Based on the information given above only, could the distribution of household water use for that county be approximately normal? Explain your answer. How many households stay within one standard deviation of the mean in terms of water usage in the county? A random sample of 125 households will be selected, and the mean water use will be calculated for the households in the sample. Is the sampling distribution of the sample mean for random samples of size 125 approximately normal if the original population is not normal? Explain. Use a theorem to justify your answer. Suppose that the annual indoor water use (in thousands of gallons) for the same county is approximately normal distributed with a mean 75 and standard deviation 20. If a random sample of 35 households is selected, what is the probability that their mean indoor water use (in thousands of gallons) will be greater than 100?Explanation / Answer
Answer:
a).
We cannot say the distribution is approximately normal by the given information.
b).
For approximately normal distribution, 68% of the data within 1 sd limits
68% of 45000 is 30600
Answer: 30600
c).
sampling distribution of sample mean of samples of 125 approximately normal even when the population is not normal because of central limit theorem.
The central limit theorem states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough.(sample size >30).
c).
standard error = sd/sqrt(n) =20/sqrt(35) =3.3806
z value for 100, z =(100-75)/3.3806=7.40
P( mean x>100) = P( z >7.40)
=0.0000.
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