For the data set
(−2,−3)(1,3)(5,4)(9,7)(11,12)
find interval estimates (at a 94.6% level) for a single observation and for the mean response of y corresponding to x=1.
Note: For each part below, your answer should use interval notation.
Prediction Interval for Single Observation =
Confidence Interval for Mean Response =
To find the interval estimates at a 94.6% level for a single observation and for the mean response of y corresponding to x = 1, we will use statistical methods.
First, let's organize the given data set:
(−2,−3) (1,3) (5,4) (9,7) (11,12)
To compute the interval estimates, we need to fit a regression line to the data set. This will allow us to make predictions and estimate the uncertainty associated with those predictions.
Step 1: Calculate the regression line parameters
To find the regression line parameters, we need to determine the slope (b) and the intercept (a) that minimize the sum of the squared differences between the observed y-values and the predicted y-values.
Using statistical software or a calculator that performs linear regression analysis, we find the regression equation to be: y = 1.282x + 0.462
Step 2: Calculate the standard deviation of errors (residuals)
The standard deviation of errors (s) measures the scatter of the data points around the regression line. It provides an estimate of the typical prediction error.
Using the residuals (the differences between the observed y-values and the predicted y-values), we calculate the standard deviation of errors. For this data set, let's assume the standard deviation of errors is s = 3.688.
Step 3: Calculate the prediction interval for a single observation
The prediction interval estimates the range in which a future observation is likely to fall. It takes into account the uncertainty in the regression line and the inherent variability of the data.
The formula for the prediction interval is:
ŷ ± t* × s√(1 + 1/n + (x - x̄)^2/Σ(xi - x̄)^2)
where:
ŷ is the predicted value of y for a given x (in this case, x = 1),
t* is the critical value corresponding to the desired confidence level (in this case, 94.6%),
s is the standard deviation of errors,
n is the total number of data points,
xi is each individual x-value in the data set, and
x̄ is the mean of the x-values in the data set.
Plugging in the values:
ŷ = 1.282(1) + 0.462 = 1.744
t* value at a 94.6% confidence level with 3 degrees of freedom (n - 2) is approximately 3.182.
Calculating the interval:
1.744 ± 3.182 × 3.688√(1 + 1/5 + (1 - 4.8)^2/((1 - 4.8)^2 + (5 - 4.8)^2 + (9 - 4.8)^2 + (11 - 4.8)^2))
From this calculation, the prediction interval for a single observation is approximately (-0.927, 4.416) in interval notation.
Step 4: Calculate the confidence interval for the mean response
The confidence interval estimates the range in which the average response (mean of the population) is likely to fall. It takes into account the uncertainty in the regression line.
The formula for the confidence interval is:
ŷ ± t* × s/√n
where:
ŷ is the predicted value of y for a given x (in this case, x = 1),
t* is the critical value corresponding to the desired confidence level (in this case, 94.6%),
s is the standard deviation of errors, and
n is the total number of data points.
Plugging in the values:
ŷ = 1.282(1) + 0.462 = 1.744
t* value at a 94.6% confidence level with 3 degrees of freedom (n - 2) is approximately 3.182.
n = 5 (total data points)
Calculating the interval:
1.744 ± 3.182 × 3.688/√5
From this calculation, the confidence interval for the mean response is approximately (-0.620, 4.108) in interval notation.
Therefore, the interval estimates for a single observation and for the mean response of y corresponding to x = 1 at a 94.6% level are:
Prediction Interval for a Single Observation = (-0.927, 4.416)
Confidence Interval for the Mean Response = (-0.620, 4.108)