A sample of 11 observations taken from a normally distributed population produced the following data.
−7.1
10.3
8.7
−3.6
−6.0
−7.5
5.2
3.7
9.8
−4.4
6.4
a) What is the point estimate of μ?
b) Make a 95% confidence interval for μ.
c) What is the margin of error of estimate for μ in part b?
a) 1.409
b) ( -2.822, 5.64)
c) 4.23
a) The point estimate of μ is the sample mean. So, let me calculate the sample mean for you...
*drumroll*
The sample mean is approximately -0.73.
b) Ah, confidence intervals, the shy cousins of point estimates. To calculate a 95% confidence interval for μ, we need to take into account the sample size and the sample mean. Give me a moment to do the math...
*taps fingers on imaginary calculator*
The 95% confidence interval for μ is approximately -4.36 to 2.90.
c) Now, let's talk about the margin of error. In the context of estimating μ, the margin of error is half the width of the confidence interval. So, let me calculate it for you...
*puts on thinking cap*
The margin of error of estimate for μ in this case is approximately 3.63.
To answer these questions, we need to follow a step-by-step approach. Let's go through each question one by one.
a) What is the point estimate of μ?
The point estimate of μ is the sample mean (X̄). We can calculate it by adding up all the observations and dividing by the total number of observations.
Sum of observations = -7.1 + 10.3 + 8.7 + -3.6 + -6.0 + -7.5 + 5.2 + 3.7 + 9.8 + -4.4 + 6.4
Sum of observations = 0.1
Number of observations (n) = 11
X̄ = Sum of observations / n
X̄ = 0.1 / 11
X̄ = 0.0091 (rounded to four decimal places)
Therefore, the point estimate of μ is approximately 0.0091.
b) Make a 95% confidence interval for μ.
To calculate the confidence interval for μ, we can use the formula:
Confidence Interval = X̄ ± (Z * (σ / √n))
Where:
X̄ = Sample mean
Z = Z-score corresponding to the desired confidence level (95% confidence level has a Z-score of 1.96)
σ = Standard deviation of the population (unknown in this case)
n = Number of observations
Since we don't have the population standard deviation (σ), we can use the sample standard deviation (s) as an estimate. We can calculate the sample standard deviation using the formula:
s = √[(Σ(xi - X̄)²) / (n - 1)]
Where:
xi = individual observation
X̄ = Sample mean
n = Number of observations
Calculating the sample standard deviation:
s = √[((-7.1 - 0.0091)² + (10.3 - 0.0091)² + (8.7 - 0.0091)² + (-3.6 - 0.0091)² + (-6.0 - 0.0091)² + (-7.5 - 0.0091)² + (5.2 - 0.0091)² + (3.7 - 0.0091)² + (9.8 - 0.0091)² + (-4.4 - 0.0091)² + (6.4 - 0.0091)²) / (11 - 1)]
s = √[679.54 / 10]
s = 7.69 (rounded to two decimal places)
Now, we can calculate the confidence interval:
Confidence Interval = X̄ ± (Z * (s / √n))
Confidence Interval = 0.0091 ± (1.96 * (7.69 / √11))
Confidence Interval = 0.0091 ± 5.9961
Confidence Interval ≈ (-5.987, 6.005)
Therefore, the 95% confidence interval for μ is approximately (-5.987, 6.005).
c) What is the margin of error of estimate for μ in part b?
The margin of error in estimate for μ is half the width of the confidence interval. We can calculate it by subtracting the lower bound from the upper bound and dividing it by 2.
Margin of Error = (Upper bound - Lower bound) / 2
Margin of Error = (6.005 - (-5.987)) / 2
Margin of Error = 11.992 / 2
Margin of Error ≈ 5.996
Therefore, the margin of error of estimate for μ is approximately 5.996.
a) The point estimate of μ (population mean) can be calculated by finding the average (mean) of the sample data. To do this, add up all the values and divide by the number of observations (11 in this case):
Sample mean = (−7.1 + 10.3 + 8.7 + −3.6 + −6.0 + −7.5 + 5.2 + 3.7 + 9.8 + −4.4 + 6.4) / 11
b) To make a 95% confidence interval for μ, we need to determine the margin of error and add/subtract it to/from the point estimate. The formula for the margin of error is:
Margin of error = critical value * (sample standard deviation / square root of sample size)
First, let's calculate the critical value for a 95% confidence level. Assuming a large sample size (11 in this case), we can use the z-distribution. The critical value can be obtained from a standard normal distribution table or calculated using statistical software. For a 95% confidence level, the critical value is approximately 1.96.
Next, we need to calculate the sample standard deviation. This can be computed by finding the square root of the sum of squared differences between each observation and the sample mean, divided by the sample size minus 1 (to correct for degrees of freedom). Here's the calculation:
Deviation from the mean = (−7.1 - sample mean)^2 + (10.3 - sample mean)^2 + (8.7 - sample mean)^2 + (−3.6 - sample mean)^2 + (−6.0 - sample mean)^2 + (−7.5 - sample mean)^2 + (5.2 - sample mean)^2 + (3.7 - sample mean)^2 + (9.8 - sample mean)^2 + (−4.4 - sample mean)^2 + (6.4 - sample mean)^2
Sample standard deviation = square root of (sum of deviations from the mean / (sample size - 1))
Once we have the critical value and the sample standard deviation, we can calculate the margin of error using the formula mentioned earlier.
c) The margin of error of estimate for μ is the range within which the true population mean is likely to fall. It is the value obtained in part b when the point estimate is added to or subtracted from the margin of error.
To calculate it, we multiply the margin of error (obtained in part b) by −1 to get the lower bound, and add the margin of error to the point estimate of μ to get the upper bound.
For example, if the point estimate of μ is 5.0 and the margin of error is 1.2, then:
Lower bound = 5.0 - 1.2
Upper bound = 5.0 + 1.2
The margin of error of estimate for μ would be (Lower bound, Upper bound).