Answer:
Step-by-step explanation:
n = 12
Mean = (4.58 + 5.72 + 4.77 + 4.76 + 5.19 + 5.05 + 4.80 + 4.77 + 4.75 + 5.02 + 4.74 + 4.56)/12 = 4.8925
Standard deviation = √(summation(x - mean)²/n
Summation(x - mean)² = (4.58 - 4.8925)^2 + (5.72 - 4.8925)^2 + (4.77 - 4.8925)^2 + (4.76 - 4.8925)^2 + (5.19 - 4.8925)^2 + (5.05 - 4.8925)^2 + (4.80 - 4.8925)^2 + (4.77 - 4.8925)^2 + (4.75 - 4.8925)^2 + (5.02 - 4.8925)^2 + (4.74 - 4.8925)^2 + (4.56 - 4.8925)^2 = 1.122225
Standard deviation = √(1.122225/12
s = 0.31
a) Point estimate = sample mean = 4.8925
Confidence interval is written in the form,
(Sample mean - margin of error, sample mean + margin of error)
Margin of error = z × s/√n
Where
From the information given, the population standard deviation is unknown and the sample size is small, hence, we would use the t distribution to find the z score
In order to use the t distribution, we would determine the degree of freedom, df for the sample.
df = n - 1 = 12 - 1 = 11
b) Since confidence level = 95% = 0.95, α = 1 - CL = 1 – 0.95 = 0.05
α/2 = 0.05/2 = 0.025
the area to the right of z0.025 is 0.025 and the area to the left of z0.025 is 1 - 0.025 = 0.975
Looking at the t distribution table,
z = 2.201
Margin of error = 2.201 × 0.31/√12
= 0.197
95% confidence interval = 4.8925 ± 0.197
Upper limit = 4.8925 + 0.197 = 5.0895
Lower limit = 4.8925 - 0.197 = 4.6955
We are 95% confident that the population mean of the rain water ph lies between 4.6955 and 5.0895
c) For 99% confidence level, z = 3.106
Margin of error = 3.106 × 0.31/√12
= 0.278
99% confidence interval = 4.8925 ± 0.278
Upper limit = 4.8925 + 0.278 = 5.1705
Lower limit = 4.8925 - 0.278 = 4.6145
We are 99% confident that the population mean of the rain water ph lies between 4.6145 and 5.1705
d) The interval gets wider as the confidence level is increased. This is logical since the test score is higher for 99% and therefore, increases the range of values. Since we want to be more confident, the range of values must be extended.
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