The following data represent the running lengths (in...

The following data represent the running lengths (in minutes) of the winners of the Academy Award for Best Picture for the years $2004-2009$ $$ \begin{array}{lc} \hline \text { 2004: } \text { Million Dollar Baby } & 132 \\ \hline \text { 2005: Crash } & 112 \\ \hline \text { 2006: The Departed } & 151 \\ \hline \text { 2007: No Country for Old Men } & 122 \\ \hline \text { 2008: Slumdog Millionaire } & 120 \\ \hline \text { 2009: The Hurt Locker } & 131 \\ \hline \end{array} $$ (a) Compute the population mean, $\mu$. (b) List all possible samples with size $n=2 .$ There should be ${ }_{6} C_{2}=15$ samples. (c) Construct a sampling distribution for the mean by listing the sample means and their corresponding probabilities. (d) Compute the mean of the sampling distribution. (e) Compute the probability that the sample mean is within 5 minutes of the population mean running time. (f) Repeat parts (b)-(e) using samples of size $n=3$. Comment on the effect of increasing the sample size.

The following data represent the running lengths (in minutes) of the winners of the Academy Award for Best Picture for the years $2004-2009$
$$
\begin{array}{lc}
\hline \text { 2004: } \text { Million Dollar Baby } & 132 \\
\hline \text { 2005: Crash } & 112 \\
\hline \text { 2006: The Departed } & 151 \\
\hline \text { 2007: No Country for Old Men } & 122 \\
\hline \text { 2008: Slumdog Millionaire } & 120 \\
\hline \text { 2009: The Hurt Locker } & 131 \\
\hline
\end{array}
$$
(a) Compute the population mean, $\mu$.
(b) List all possible samples with size $n=2 .$ There should be ${ }_{6} C_{2}=15$ samples.
(c) Construct a sampling distribution for the mean by listing the sample means and their corresponding probabilities.
(d) Compute the mean of the sampling distribution.
(e) Compute the probability that the sample mean is within 5 minutes of the population mean running time.
(f) Repeat parts (b)-(e) using samples of size $n=3$. Comment on the effect of increasing the sample size.

Key Concepts

Population Mean

In statistics, the population mean is the average of a set of values in the entire population under study. It is a fixed value (often denoted as ?) and is computed by summing all observations in the population and dividing by the number of observations. This concept is fundamental for comparing estimates derived from samples and for assessing the accuracy of sample-based estimates.

Sample Mean

The sample mean is the average of observations taken from a subset (sample) of the larger population. It serves as an estimate of the population mean and is computed by summing the values in the sample and dividing by the sample size. The sample mean is a random variable since it varies from sample to sample, and its study is central to understanding statistical estimation and inference.

Sampling Distribution

The sampling distribution is the probability distribution of a given statistic, such as the sample mean, derived from a large number of samples drawn from the same population. It describes how the sample mean varies with different samples and is key to making inferences about the population. The concept underlies many statistical methods, including hypothesis testing and the construction of confidence intervals.

Combinatorics in Sampling

Combinatorics in sampling involves counting the number of ways samples of a certain size can be drawn from the population. This is typically done using combination formulas, such as nCk, to list all possible samples. Understanding this concept is essential when constructing the sampling distribution explicitly by considering all possible samples and their associated probabilities.

Effect of Sample Size

The sample size affects the variability of the sample mean; as the sample size increases, the variability (or standard error) of the sample mean typically decreases. This means that with larger samples, the sampling distribution of the mean becomes more concentrated around the true population mean, improving the precision of statistical estimates. This principle is a key aspect of the law of large numbers and is vital when designing studies and experiments.

Transcript

Please give Ace some feedback