A random sample of 150 towns in a western state had a mean annual precipitation of 2.86 inches. Assume that the population standard deviation is known to be 0.78 inches. Compute the 95% confidence interval for μ.
95% = mean ± 1.96 SEm
SEm = SD/√n
I'll let you do the calculations.
To compute the 95% confidence interval for the population mean (μ) based on the given information, we can use the formula:
confidence interval = sample mean ± (critical value * standard error)
Step 1: Find the critical value.
Since we want a 95% confidence interval, we need to find the critical value corresponding to a 95% confidence level. The standard practice is to use a Z-table or Z-score calculator. For a 95% confidence level, the critical value is approximately 1.96.
Step 2: Find the standard error.
The standard error (SE) is a measure of how much the sample mean is likely to vary from the true population mean. It is calculated using the formula:
standard error = population standard deviation / sqrt(sample size)
Here, the population standard deviation (σ) is given as 0.78 inches, and the sample size (n) is 150.
standard error = 0.78 / sqrt(150)
Step 3: Calculate the confidence interval.
The confidence interval can now be calculated by substituting the values into the formula:
confidence interval = sample mean ± (critical value * standard error)
sample mean = 2.86 inches
critical value = 1.96 (for a 95% confidence level)
standard error ≈ 0.078 inches (computed in Step 2)
Using these values:
confidence interval = 2.86 ± (1.96 * 0.078)
Simplifying:
confidence interval = 2.86 ± 0.153
The 95% confidence interval for μ is approximately (2.707, 3.013) inches.
To compute the 95% confidence interval for μ, we can use the formula:
Confidence Interval = (sample mean) ± (critical value) * (standard deviation / √sample size)
1. Start by determining the critical value. Since we want a 95% confidence interval, we need to find the corresponding z-value. The z-value can be obtained from a standard normal distribution table or using a calculator or software. For a 95% confidence interval, the z-value is 1.96.
2. Next, plug in the given values into the formula:
Sample mean = 2.86 inches
Standard deviation = 0.78 inches
Sample size = 150
Confidence Interval = 2.86 ± 1.96 * (0.78 / √150)
3. Simplify the expression:
Confidence Interval = 2.86 ± 1.96 * (0.78 / 12.25)
4. Calculate the standard error:
Standard error = 0.78 / √150 ≈ 0.0635
5. Plug in the standard error value into the formula:
Confidence Interval = 2.86 ± 1.96 * 0.0635
6. Calculate the upper and lower limits of the confidence interval:
Upper Limit = 2.86 + 1.96 * 0.0635 ≈ 3.19 inches
Lower Limit = 2.86 - 1.96 * 0.0635 ≈ 2.53 inches
Therefore, the 95% confidence interval for μ is approximately (2.53 inches, 3.19 inches).