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 $1999-2004$ $$\begin{array}{lr} \text { 2004: Million Dollar Baby } & 132 \\ \hline \text { 2003: The Lord of the Rings: The Return of the King } & 201 \\ \hline \text { 2002: Chicago } & 112 \\ \hline \text { 2001: A Beautiful Mind } & 134 \\ \hline \text { 2000: Gladiator } & 155 \\ \hline \text { 1999: American Beauity } & 120 \end{array}$$ (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 15 minutes of the population mean running times. (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 $1999-2004$
$$\begin{array}{lr}
\text { 2004: Million Dollar Baby } & 132 \\
\hline \text { 2003: The Lord of the Rings: The Return of the King } & 201 \\
\hline \text { 2002: Chicago } & 112 \\
\hline \text { 2001: A Beautiful Mind } & 134 \\
\hline \text { 2000: Gladiator } & 155 \\
\hline \text { 1999: American Beauity } & 120
\end{array}$$
(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 15 minutes of the population mean running times.
(f) Repeat parts $(b)-(e)$ using samples of size $n=3$ Comment on the effect of increasing the sample size.

Key Concepts

Population Mean

The population mean is the average of all values in an entire population. It is calculated by summing all individual measurements and then dividing by the total number of measurements. This parameter characterizes the central tendency of the population and provides a baseline for comparing sample statistics.

Sample Mean

The sample mean is the average computed from a subset (sample) drawn from the population. It serves as an estimate of the population mean. Because each sample might yield different values, the sample mean is considered a random variable that varies from sample to sample.

Sampling Distribution

The sampling distribution is the probability distribution of a given statistic (such as the sample mean) over all possible samples of a fixed size drawn from the population. It provides critical insight into the variability and reliability of the sample statistic as an estimator of the population parameter.

Combinatorics in Sampling

Combinatorics involves counting and enumerating the possible ways samples can be drawn from a population. Concepts such as combinations, which determine the number of unique samples that can be selected without regard to order, are essential here. For example, when selecting a sample of size two from six values, the number of different samples is given by the combination formula ‘6 choose 2’.

Probability Calculation for Sample Means

This concept involves determining the likelihood of obtaining a particular sample mean when sampling from the population. It includes calculating the probabilities associated with each possible sample mean, often using the frequencies of occurrence in the sampling distribution. This helps in assessing the precision and variability of the sample mean in relation to the population mean.

Effect of Sample Size on Sampling Distribution

Increasing the sample size generally leads to a sampling distribution that is less variable and more concentrated around the population mean. This reduction in variability makes the sample mean a more reliable estimator. The central limit theorem supports this by stating that, for sufficiently large sample sizes, the sampling distribution of the sample mean approximates a normal distribution, regardless of the shape of the population distribution.

Transcript

00:01 Alright.

00:06 It's looking at movies.

00:07 Academy award winning best pictures for a six -year period and they're running lengths and we're looking for the population mean of those running lengths.

00:16 So there they are listed as the titles and the list and the average of them using the average function in the spreadsheet.

00:26 We get that value of 142 and a third.

00:29 Then list possible samples of size n equals two.

00:32 So we get them right here and nice.

00:34 Just went down there's the list here but what we're going to be doing with them is find the average of those size two samples so that here are the there's the harris how i set up i looked for the million dollar baby and then lord of the rings and then just went down the line and shifted that lord of the rings sell down to chicago and the beautiful mine gladiator american beauty and then i said all right no more million million dollar baby lord of the rings and all the other combinations.

01:06 All right, no more lord of the rings.

01:09 We'll do chicago.

01:10 And going down the line.

01:12 All right, no more chicago.

01:13 Beautiful mind.

01:15 And then, there we go.

01:16 Then the last ones.

01:18 And you get this nice triangle because 15 is a triangular number.

01:22 It's kind of fun.

01:23 And then you take those and you list them out, sort them.

01:26 And there's nothing that repeats.

01:28 So it's all one or all unique.

01:31 So it's one out of 15.

01:35 And then we're asked to determine of the mean of the sampling distribution which ends up being 142 and a third which is the population mean which is kind of cool right because if we do equals average and go up to it and look at that kind of neat and we're going to compute the probability that the sample means within 15 so i'm going to write this out because this is easy nice to see we get the probability that the mean is less than 15 minutes or more than 15 minutes of the mean.

02:12 The mean being 142 and a third minus 15 is less than x bar which is less than 142 in a third plus 15 which ends up being the probability of 127 in a third less than x bar which is less than 157 and a third.

02:41 And so when i initially did this i way over thought it.

02:44 I was doing stuff with z scores and all this stuff standard error the mean, which would work.

02:50 However, we don't know anything about the population, so we can't do that...

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