Answer:
Option B is correct.
It is reasonable to believe that the mean magnesium ion concentration may be greater than 199.5 as the confidence interval obtained contains values that are greater than 199.5
Explanation:
Complete Question
Six measurements were made of the magnesium ion concentration (in parts per million, or ppm) in a city's municipal water supply, with the following results. It is reasonable to assume that the population is approximately normal.
170 201 199 202 173 153
Based on a 95% confidence interval for the mean magnesium ion concentration, is it reasonable to believe that the mean magnesium ion concentration may be greater than 199.5? (Hint: you should first calculate the 95% confidence interval for the mean magnesium ion concentration.)
A) The likelihood cannot be determined.
B) Yes
C) No
Solution
For this question, obtaining the confidence interval will give a clear solution to the problem.
Since the Confidence Interval for the population mean is basically an interval of range of values where the true population mean can be found with a certain level of confidence, if the range obtained contains values greater than the standard we are comparing against (199.5), then the confidence interval proves that the mean magnesium ion may be greater than 199.5.
But to obtain the confidence interval, we need the mean and standard deviation for the sample.
170, 201, 199, 202, 173, 153
Mean = (sum of variables)/(total number of variables)
Sum of variables = 170+201+199+202+173+153 = 1098
Total number of variables = 6
Mean = (1098/6) = 183
Standard deviation = σ = √[Σ(x - xbar)²/N]
x = each variable
xbar = mean = 183
N = number of variables = 6
Σ(x - xbar)² = (170-183)² + (201-183)² + (199-183)² + (202-183)² + (173-183)² + (153-183)²
= 169 + 324 + 256 + 361 + 100 + 900
= 2110
σ = √(2110/6) = 18.75
Mathematically,
Confidence Interval = (Sample mean) ± (Margin of error)
Sample Mean = 183
Margin of Error is the width of the confidence interval about the mean.
It is given mathematically as,
Margin of Error = (Critical value) × (standard Error of the mean)
Critical value will be obtained using the t-distribution. This is because there is no information provided for the population mean and standard deviation.
To find the critical value from the t-tables, we first find the degree of freedom and the significance level.
Degree of freedom = df = n - 1 = 6 - 1 = 5.
Significance level for 95% confidence interval
(100% - 95%)/2 = 2.5% = 0.025
t (0.025, 5) = 2.57 (from the t-tables)
Standard error of the mean = σₓ = (σ/√n)
σ = standard deviation of the sample = 18.75
n = sample size = 6
σₓ = (18.75/√6) = 7.656
95% Confidence Interval = (Sample mean) ± [(Critical value) × (standard Error of the mean)]
CI = 183 ± (2.57 × 7.656)
CI = 183 ± 19.675
95% CI = (163.325, 202.675)
95% Confidence interval = (163.3, 202.7)
It is reasonable to believe that the mean magnesium ion concentration may be greater than 199.5 as the confidence interval obtained contains values that are greater than 199.5
Hope this Helps!!!