Question

Question 1. The pH of 20 randomly selected lakes is measured. Their average pH is 5.7. Part A. Historically the standard deviation in the pH values is 0.9. Use this standard deviation for the following questions. Part Ai. Build a 95% confidence interval for the population mean lake pH. Part Ali. Build a 90% confidence interval for the population mean lake pH. Part Alli. Build an 80% confidence interval for the population mean lake pH. Part Aiv. Compare the intervals you created in Ai, Aii and Aiii. What effect does changing the level of confidence have on the interval? Part Av. Test whether the population mean pH is more than 6. Part Avi. To be accurate to within 0.1 with 95% confidence, how many lakes should we measure? Part B. For the 20 measured lakes the standard deviation in the pH values is 0.9. Use this standard deviation for the following questions. Part Bi. Build a 95% confidence interval for the population mean lake pH. Part Bii. Compare the confidence intervals in Bi and Ai. What effect does not knowing the value of o have on the interval? Part Biii. Test whether the population mean pH differs from 6. Part C. Five of the 20 lakes have a pH less than 4.5. Part Ci. Build a 95% confidence interval for the proportion of lakes that have a pH less than 4.5. Part Cii. Test whether the population proportion of pH's that are less than 4.5 differs from 0.5. Part Ciii. To be accurate to within 0.1 with 95% confidence, how many lakes should we measure?

Answer 1

i) M = 5.7
Z critical = 1.96
s_M = √(0.9²/20) = 0.2

μ = M ± Z(s_M)
μ = 5.7 ± 1.96*0.2
μ = 5.7 ± 0.392

95% CI [5.308, 6.092].

You can be 95% confident that the population mean (μ) falls between 5.308 and 6.092.

ii) Z critical = 1.64
sM = √(0.9^2/20) = 0.2

μ = M ± Z(sM)
μ = 5.7 ± 1.64*0.2
μ = 5.7 ± 0.328
90% CI [5.372, 6.028].

You can be 90% confident that the population mean (μ) falls between 5.372 and 6.028.

iii) Z critical = 1.28
sM = √(0.9^2/20) = 0.2

μ = M ± Z(sM)
μ = 5.7 ± 1.28*0.2
μ = 5.7 ± 0.256
80% CI [5.444, 5.956].

You can be 80% confident that the population mean (μ) falls between 5.444and 5.956

iv) The width increases as the confidence level increases. Confidence level changing the width of confidence interval.

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