Final answer:
To decide if the mean magnesium ion concentration might be greater than 196 ppm, calculate the 90% confidence interval using the sample mean and standard deviation and then check if the upper bound of the interval exceeds 196 ppm.
Step-by-step explanation:
To determine if it is reasonable to believe that the mean magnesium ion concentration may be greater than 196 ppm given a 90% confidence interval for the mean, we need to calculate the confidence interval from the provided data. Here are the six measurements given: 179, 168, 192, 125, 174, and 200 ppm. First, we calculate the sample mean (x) and the sample standard deviation (s).
The sample mean is:
x = (179 + 168 + 192 + 125 + 174 + 200) / 6
The sample standard deviation is calculated using the formula:
s = sqrt(Σ(xi - x)² / (n - 1))
Where Σ represents the sum, xi is each individual measurement, and n is the number of measurements.
After calculating the mean and standard deviation, we use the t-distribution (since the sample size is small and we do not know the population standard deviation) to calculate the 90% confidence interval (CI) for the mean. The formula is:
CI = x ± t*(s/√n)
Where t* is the t-value from the t-distribution table corresponding to the desired confidence level and degrees of freedom (n-1), and √n is the square root of the sample size.
Once the CI is calculated, we can look at the upper bound of the interval. If the upper bound is less than 196 ppm, it would not be reasonable to believe that the mean concentration could be greater than 196 ppm. If the upper bound is greater than or equal to 196 ppm, then it may be reasonable.