Students took n = 35 samples of water from the east basin of Lake
Macatawa and measured the amount of sodium in parts per million. For their data,
they calculated notation bar x= 24.11 and s^2= 24.44. Find an approximate 90% confidence interval for μ , the mean of the amount of sodium in parts per million.
90% = mean ± 1.645 SEm (Standard Error of the mean)
SEm = s/√(n-1)
I'll let you do the calculations.
To find the approximate 90% confidence interval for the mean (μ) of the amount of sodium in parts per million, we can use the t-distribution.
Given:
Sample mean (x̄) = 24.11
Sample variance (s^2) = 24.44
Sample size (n) = 35
Step 1: Find the standard error (SE)
SE = √(s^2 / n)
= √(24.44 / 35)
≈ 0.816
Step 2: Find the critical value from the t-distribution.
Since the sample size is 35, the degrees of freedom (df) is (n-1) = (35-1) = 34.
For a 90% confidence level, alpha (α) = 1 - 0.90 = 0.10
Using a t-table or a t-distribution calculator with df = 34 and α/2 = 0.10/2 = 0.05, the critical value is approximately 1.691.
Step 3: Calculate the margin of error (ME)
ME = Critical value * SE
≈ 1.691 * 0.816
≈ 1.381
Step 4: Calculate the confidence interval.
Confidence interval = (x̄ - ME, x̄ + ME)
= (24.11 - 1.381, 24.11 + 1.381)
≈ (22.729, 25.491)
Approximately, the 90% confidence interval for μ, the mean of the amount of sodium in parts per million, is (22.729, 25.491).
To find an approximate 90% confidence interval for the mean, μ, of the amount of sodium in parts per million, we can use the t-distribution.
Here's how you can calculate it step by step:
Step 1: Determine the sample size, n, which is given as 35.
Step 2: Calculate the sample mean, notation x, which is given as 24.11.
Step 3: Calculate the sample standard deviation, s, by taking the square root of the sample variance, s^2. In this case, s^2 is given as 24.44, so s = √(24.44) ≈ 4.94.
Step 4: Determine the confidence level, which is given as 90%. This means we need to find the critical value for a two-tailed test with a confidence level of 90%. The degrees of freedom for a sample size of 35 is n-1 = 35-1 = 34. Using a t-table or calculator, the critical value for a 90% confidence level with 34 degrees of freedom is approximately 1.69.
Step 5: Calculate the margin of error, which is given by the formula:
Margin of Error = critical value * (standard deviation / √sample size)
In this case, the critical value is 1.69, the standard deviation is 4.94, and the sample size is 35. Therefore, the margin of error is:
Margin of Error = 1.69 * (4.94 / √35) ≈ 1.61
Step 6: Calculate the lower and upper bounds of the confidence interval.
Lower Bound = sample mean - margin of error
Upper Bound = sample mean + margin of error
In this case, the lower bound is 24.11 - 1.61 ≈ 22.50, and the upper bound is 24.11 + 1.61 ≈ 25.72.
Thus, the approximate 90% confidence interval for the mean, μ, of the amount of sodium in parts per million is (22.50, 25.72).