Question

1) A random sample of 13 lunch orders at Noodles and Company showed a mean bill of $10.32 with a standard deviation of $4.62. Find the 99 percent confidence interval for the mean bill of all lunch orders. (Round your answers to 4 decimal places.) The 99% confidence interval is from

2)

The Environmental Protection Agency (EPA) requires that cities monitor over 80 contaminants in their drinking water. Samples from the Lake Huron Water Treatment Plant gave the results shown here. Only the range is reported, not the mean (presumably the mean would be the midrange).

  

Substance	MCLG Range Detected	Allowable MCLG	Origin of Substance
Chromium	0.43 to 0.64	100	Discharge from steel and pulp mills, natural erosion
Barium	0.004 to 0.019	2	Discharge from drilling wastes, metal refineries, natural erosion
Fluoride	1.06 to 1.13	4	Natural erosion, water additive, discharge from fertilizer and aluminum factories
MCLG = Maximum contaminant level goal

For each substance, estimate the standard deviation σ by assuming uniform distribution and normal distribution shown in Table 8.11 in Section 8.8. (Round your answers to 4 decimal places.)

  

	Uniform Distribution	Normal Distribution
Chromium
Barium
Fluoride

Answer 1

Answer 1

• Sample statistic. Since we are trying to estimate a population mean, we choose the sample mean M ($10.32) as the sample statistic.
• We have sample of size 13
• Sample stander deviation is given as $4.62.
• Then σ_M is the standard error of the mean would be
• Since we need to find 99% confidence interval, here = 1 - (99/100) = 0.01
• For =0.01,form normal distribution we have value got Z_0.01 = 2.58

• Now the 99% confidence interval for the mean with the following formula:

  Lower limit = M - Z_.99σ_M
  
  Upper limit = M + Z_.99σ_M

• Hence the required confidence interval for given data is

Lower limit = M - Z_.99σ_M

=$10.32 - 2.58 * 1.2814

= 7.0141

Upper limit = M + Z_.99σ_M

=$10.32 + 2.58 * 1.2814

= 13.6259

Answer 2

Suppose X follows uniform distribution with range values a to b, then variance of X would be

Analysing the loss function with respect to optimal Gaussians we can fit U(a,b) to N(μ,σ) we would obtain:

so this minimization problem corresponds to a linear re-scaling of the uniform parameters in terms of μ and σ.

Since mean can be assumed by the midrange, it would be μ = (a+b)/2 and by solving above equation we will get σ.

Uniform Distribution
Substance	Range	a	b	Variance	standard deviation
Chromium	0.43 to 0.64	0.43	0.64	0.0037	0.0606
Barium	0.004 to 0.019	0.004	0.019	0.0000	0.0043
Fluoride	1.06 to 1.13	1.06	1.13	0.0004	0.0202

Normal Distribution
Substance	Range	a	b	Mean (μ)	standard deviation(σ)
Chromium	0.43 to 0.64	0.43	0.64	0.5350	0.0350
Barium	0.004 to 0.019	0.004	0.019	0.0115	0.0025
Fluoride	1.06 to 1.13	1.06	1.13	1.0950	0.0117

Hence from these tow tables we can write

Substance	Uniform distribution	Normal Distribution
Chromium	0.0606	0.0350
Barium	0.0043	0.0025
Fluoride	0.0202	0.0117