Following are the published weights (in pounds) of all of...

Following are the published weights (in pounds) of all of the team members of the San Francisco 49ers from a previous year. 177; 205; 210; 210; 232; 205; 185; 185; 178; 210; 206; 212; 184; 174; 185; 242; 188; 212; 215; 247; 241; 223; 220; 260; 245; 259; 278; 270; 280; 295; 275; 285; 290; 272; 273; 280; 285; 286; 200; 215; 185; 230; 250; 241; 190; 260; 250; 302; 265; 290; 276; 228; 265 a. Organize the data from smallest to largest value. b. Find the median. c. Find the first quartile. d. Find the third quartile. e. Construct a box plot of the data. f. The middle 50$\%$ of the weights are from ____ to ____ . g. If our population were all professional football players, would the above data be a sample of weights or the population of weights? Why? h. If our population included every team member who ever played for the San Francisco 49ers, would the above data be a sample of weights or the population of weights? Why? i. Assume the population was the San Francisco 49ers. Find: $$\begin{array}{l}{\text { i. the population mean, } \mu} \\ {\text { ii. the population standard deviation, } \sigma \text { . }} \\ {\text { iii. the weight that is two standard deviations below the mean. }} \\ {\text { iv. When Steve Young, quarterback, played football, he weighed 205 pounds. How many standard }} \\ {\text { deviations above or below the mean was he? }}\end{array}$$ j. That same year, the mean weight for the Dallas Cowboys was 240.08 pounds with a standard deviation of 44.38 pounds. Emmit Smith weighed in at 209 pounds. With respect to his team, who was lighter, Smith or Young? How did you determine your answer?

Following are the published weights (in pounds) of all of the team members of the San Francisco 49ers from a previous year.
177; 205; 210; 210; 232; 205; 185; 185; 178; 210; 206; 212; 184; 174; 185; 242; 188; 212; 215; 247; 241; 223; 220; 260; 245; 259; 278; 270; 280; 295; 275; 285; 290; 272; 273; 280; 285; 286; 200; 215; 185; 230; 250; 241; 190; 260; 250; 302; 265; 290; 276; 228; 265
a. Organize the data from smallest to largest value.
b. Find the median.
c. Find the first quartile.
d. Find the third quartile.
e. Construct a box plot of the data.
f. The middle 50$\%$ of the weights are from ____ to ____ .
g. If our population were all professional football players, would the above data be a sample of weights or the population of weights? Why?
h. If our population included every team member who ever played for the San Francisco 49ers, would the above data be a sample of weights or the population of weights? Why?
i. Assume the population was the San Francisco 49ers. Find:
$$\begin{array}{l}{\text { i. the population mean, } \mu} \\ {\text { ii. the population standard deviation, } \sigma \text { . }} \\ {\text { iii. the weight that is two standard deviations below the mean. }} \\ {\text { iv. When Steve Young, quarterback, played football, he weighed 205 pounds. How many standard }} \\ {\text { deviations above or below the mean was he? }}\end{array}$$
j. That same year, the mean weight for the Dallas Cowboys was 240.08 pounds with a standard deviation of 44.38 pounds. Emmit Smith weighed in at 209 pounds. With respect to his team, who was lighter, Smith or Young? How did you determine your answer?

Key Concepts

Standard Scores (z-scores)

A standard score, or z-score, measures how many standard deviations an individual data point is from the mean. It is used to compare values from different data sets or distributions and helps in identifying the relative position of a data point within its distribution.

Population Mean and Standard Deviation

The population mean is the average of all the observations in a complete data set, and the population standard deviation is a measure of the spread or variability of the data around the mean. These parameters are crucial for understanding the overall characteristics of a population and for conducting further statistical analysis.

Sample vs. Population

The distinction between a sample and a population is fundamental in statistics. A population includes all members of a specified group, while a sample is a subset selected from that group. The choice between using sample or population data affects the calculations and the inference about the entire group.

Box Plot

A box plot is a graphical representation of data that displays the median, quartiles, and potential outliers. It provides a visual summary of the distribution, the spread of the data, and any asymmetries or outliers present in the dataset.

Quartiles

Quartiles are values that divide a dataset into four equal parts after it has been sorted. The first quartile (Q1) marks the 25th percentile, the second quartile (Q2) is the median (50th percentile), and the third quartile (Q3) marks the 75th percentile. They are useful for assessing the spread and skewness of the data.

Median

The median is a measure of central tendency that represents the middle value in an ordered dataset. When the data is sorted, the median divides the dataset into two equal halves, which makes it useful for understanding the central location of the data, especially in skewed distributions.

Sorting Data

Sorting data involves arranging a set of values in ascending or descending order. This practice is essential in statistics to identify the median, quartiles, and range, and it facilitates further analysis by organizing the data in a structured way.

Transcript

00:02 All right, in number 115, we're looking at the weights of san francisco 49ers players from our previous year.

00:06 We have 53 different weights.

00:09 And part a says, organize the data from smallest to largest.

00:12 In order to avoid having to write these out on your paper from smallest to largest, you can use your ti 83 or 84 calculator to list these in your stat list.

00:22 And then you can make it sort it from a to z, which essentially ranks them from least to greatest.

00:29 And that's what i did.

00:31 In part b, we want to find the median.

00:33 So looking at your list, you can just find that center value, or you could look in your tia -83 or a t -i -84 calculator to figure out that your mean is 241.

00:43 Now, in order to do this, you can use that same stat list, and we'll go back to stat calculate, and then one variable statistics will calculate mean, median.

00:55 It'll calculate your five -number summary of your list.

00:58 If you insert it in list one, you just insert list one as your list, leave everything else blank and press calculate, and you can get median.

01:05 You can also get other values like your q1, which is quartile 1.

01:10 Now, quartile 1 is essentially the median of the first half of your data.

01:14 So if you look at the minimum value all the way up to your median, you have your first half of the data.

01:21 Find the median of that first half of your data, and you find quartile 1.

01:24 It's the point that separates the 25 % mark of your data from, the rest of year.

01:29 So quartile 1 is 205 .5.

01:32 And d, we weren't looking for quartile 3.

01:35 Cortal 3 is 272 .5.

01:37 Again, that's the median of the second half of your data.

01:41 E, we want to construct a box plot of the data.

01:44 So in order to do that, we need our minimum and our maximum.

01:47 We're just listing those as 174 and 302.

01:49 We can see that once we list their data in order.

01:52 And the box plot, we need a number line.

01:54 So we're just going to throw in 170 because that's close to our minimum all the way up to 300 and just to show ease i'm counting my 20s here.

02:03 So we need our five numbers above, which we call our five number summary, starting with our minimum, going to put 174 on the dot plot.

02:11 Then i'm going to throw in q1, median, q3, and max.

02:18 Now notice these dots on this plot are going to create ultimately what's going to turn into a box plot.

02:22 I just wanted you to see where these locations are going to be.

02:25 Now we're going to create a box from q1 to q3 and we're going to draw a line for our median and then we're going to draw whiskers out to our minimum and our maximum values.

02:34 And now you can see that this is the weight of all of these players on the 49ers from the previous year.

02:40 We have a pretty symmetrical distribution here.

02:42 And you can discuss it, talk about your socks, your shape, outlier, center spread, but that was not asked for in this problem.

02:49 So let's look at f.

02:50 F, what is the middle 50 % of the weight? so this is essentially your iqr, which is another way to say this, q3, minus q1.

02:58 So subtracting q3 minus q1, we get the middle 50 % to be that range of 67...

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