A random sample of eight homes in a particular suburb had...

A random sample of eight homes in a particular suburb had the following selling prices (in thousands of dollars): $\begin{array}{llllllll}192 & 183 & 312 & 227 & 309 & 396 & 402 & 390\end{array}$ a. Check for evidence of nonnormality. b. Find a point estimate of the population mean that is unbiased and efficient. c. Use an unbiased estimation procedure to find a point estimate of the variance of the sample mean. d. Use an unbiased estimator to estimate the proportion of homes in this suburb selling for less than $$\$ 250,000$$.

A random sample of eight homes in a particular suburb had the following selling prices (in thousands of dollars):
$\begin{array}{llllllll}192 & 183 & 312 & 227 & 309 & 396 & 402 & 390\end{array}$
a. Check for evidence of nonnormality.
b. Find a point estimate of the population mean that is unbiased and efficient.
c. Use an unbiased estimation procedure to find a point estimate of the variance of the sample mean.
d. Use an unbiased estimator to estimate the proportion of homes in this suburb selling for less than $$\$ 250,000$$.

Key Concepts

Assessment of Normality

This concept involves determining whether a data set conforms to a normal distribution, which is an important assumption in many statistical methods. Techniques for assessing normality include visual tools such as histograms and Q?Q plots, as well as formal tests like the Shapiro-Wilk or Anderson-Darling tests, all of which help decide whether parametric methods are appropriate.

Unbiased Estimation

Unbiased estimation refers to a property of an estimator in which its expected value equals the true parameter it estimates. This means that, on average, the estimator neither overestimates nor underestimates the parameter, making it a reliable choice in inferential statistics.

Point Estimation of the Mean

The point estimation of the mean involves using a single value as the estimate for the population mean, with the sample mean being a common choice due to its unbiasedness and efficiency. This estimator is widely used because, when the data are independent and identically distributed, its expected value equals the true mean and its variance is generally minimal.

Unbiased Estimation of the Variance of the Sample Mean

This concept focuses on estimating the variability of the sample mean without bias. It involves using the sample variance computed with a correction factor (typically dividing by n-1) and then adjusting it by the sample size (dividing by n) to provide an unbiased measure of the dispersion of the sampling distribution of the mean.

Estimation of Proportions

Proportion estimation is used when dealing with binary outcomes, where each observation can be classified into one of two categories. The sample proportion, which is the count of successes divided by the total number of observations, serves as an unbiased estimator for the true population proportion.

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