Question

Data on IQ for 33 first graders are given as follows: 82, 96, 99, 102, 103, 103, 106, 107, 108, 108, 108, 108, 109, 110, 110, 111, 113, 113, 113, 113, 115, 115, 118, 118, 119, 121, 122, 122, 127, 132, 136, 140, 146.
Calculate a point estimate of the mean value of IQ for the conceptual population of all first graders in this school, and slate which estimator you used.

Answer 1

Answer:

$\bar X = \frac{\sum_{i=1}^n x_i}{n}$

Where n represent the sample size, on this case n =33. If we use this formula we got:

$\bar X = 113.727$

And for this case we know that is the best estimator since is an unbiased estimator for the true population mean since we have this:

$E(\bar X) = E(\frac{\sum_{i=1}^n x_i}{n})= \frac{1}{n} \sum_{i=1}^n E(X_i) = \frac{n\mu}{n} = \mu$

Step-by-step explanation:

For this case we have the following values given:

82, 96, 99, 102, 103, 103, 106, 107, 108, 108, 108, 108, 109, 110, 110, 111, 113, 113, 113, 113, 115, 115, 118, 118, 119, 121, 122, 122, 127, 132, 136, 140, 146

And we want to estimate the mean value of IQ for the conceptual population.

For this case we can use as estimator for the population mean the sample mean. We know that the sample mean is given by this formula:

$\bar X = \frac{\sum_{i=1}^n x_i}{n}$

Where n represent the sample size, on this case n =33. If we use this formula we got:

$\bar X = 113.727$

And for this case we know that is the best estimator since is an unbiased estimator for the true population mean since we have this:

$E(\bar X) = E(\frac{\sum_{i=1}^n x_i}{n})= \frac{1}{n} \sum_{i=1}^n E(X_i) = \frac{n\mu}{n} = \mu$