Math / Data & Statistics

QuestionMr. Hurst, a track coach, recorded the number of runners on all the nearby track teams. \begin{tabular}{|c|c|} \hline -1) ) Number of members & ()) \\ \hline 53 & 3 \\ \hline 91 & 4 \\ \hline 104 & 3 \\ \hline \end{tabular} $X$ is the number of members that a randomly chosen team has. What is the standard deviation of $X$ ?
Round your answer to the nearest hundredth. $\square$ Submit Work it out Not feelina readv vet? This can held:

Studdy Solution

STEP 1

What is this asking? If we randomly pick a track team, how spread out is the data of team sizes? Watch out! Don't forget to weigh the number of members by the number of teams with that many members.
It's not just three data points!

STEP 2

1. Calculate the mean.
2. Calculate the variance.
3. Calculate the standard deviation.

STEP 3

First, let's find the total number of teams!
We have $3$ teams with $53$ members, $4$ teams with $91$ members, and $3$ teams with $104$ members.
So, we have a total of $3 + 4 + 3 = 10$ teams!

STEP 4

Now, let's find the total number of runners across all the teams.
There are $3 \cdot 53 = 159$ runners on the smaller teams, $4 \cdot 91 = 364$ runners on the medium-sized teams, and $3 \cdot 104 = 312$ runners on the larger teams.
That's $159 + 364 + 312 = 835$ runners in total!

STEP 5

To calculate the mean, $\mu$ , we'll divide the total number of runners by the total number of teams.
So, $\mu = \frac{835}{10} = 83.5$ runners per team on average.

STEP 6

The variance, $\sigma^2$ , measures how spread out the data is.
For each team size, we'll find the squared difference from the mean, multiply by the number of teams of that size, and then add those up.

STEP 7

For the teams with $53$ members, the squared difference is $(53 - 83.5)^2 = (-30.5)^2 = 930.25$ .
Since there are $3$ such teams, we multiply by $3$ to get $3 \cdot 930.25 = 2790.75$ .

STEP 8

For the teams with $91$ members, the squared difference is $(91 - 83.5)^2 = (7.5)^2 = 56.25$ .
Since there are $4$ such teams, we multiply by $4$ to get $4 \cdot 56.25 = 225$ .

STEP 9

For the teams with $104$ members, the squared difference is $(104 - 83.5)^2 = (20.5)^2 = 420.25$ .
Since there are $3$ such teams, we multiply by $3$ to get $3 \cdot 420.25 = 1260.75$ .

STEP 10

Adding those weighted squared differences, we get $2790.75 + 225 + 1260.75 = 4276.5$ .
To find the variance, we divide this sum by the total number of teams, which is $10$ .
So, $\sigma^2 = \frac{4276.5}{10} = 427.65$ .

STEP 11

The standard deviation, $\sigma$ , is just the square root of the variance.
So, $\sigma = \sqrt{427.65} \approx 20.68$ .
Rounding to the nearest hundredth gives us $\sigma \approx 20.68$ .

STEP 12

The standard deviation of the number of members in a randomly chosen team is approximately $20.68$ .

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Studdy Solution

STEP 1

What is this asking? If we randomly pick a track team, how spread out is the data of team sizes? Watch out! Don't forget to weigh the number of members by the number of teams with that many members.
It's not just three data points!

STEP 2

1. Calculate the mean.
2. Calculate the variance.
3. Calculate the standard deviation.

STEP 3

First, let's find the total number of teams!
We have $3$ teams with $53$ members, $4$ teams with $91$ members, and $3$ teams with $104$ members.
So, we have a total of $3 + 4 + 3 = 10$ teams!

STEP 4

Now, let's find the total number of runners across all the teams.
There are $3 \cdot 53 = 159$ runners on the smaller teams, $4 \cdot 91 = 364$ runners on the medium-sized teams, and $3 \cdot 104 = 312$ runners on the larger teams.
That's $159 + 364 + 312 = 835$ runners in total!

STEP 5

To calculate the mean, $\mu$ , we'll divide the total number of runners by the total number of teams.
So, $\mu = \frac{835}{10} = 83.5$ runners per team on average.

STEP 6

The variance, $\sigma^2$ , measures how spread out the data is.
For each team size, we'll find the squared difference from the mean, multiply by the number of teams of that size, and then add those up.

STEP 7

For the teams with $53$ members, the squared difference is $(53 - 83.5)^2 = (-30.5)^2 = 930.25$ .
Since there are $3$ such teams, we multiply by $3$ to get $3 \cdot 930.25 = 2790.75$ .

STEP 8

For the teams with $91$ members, the squared difference is $(91 - 83.5)^2 = (7.5)^2 = 56.25$ .
Since there are $4$ such teams, we multiply by $4$ to get $4 \cdot 56.25 = 225$ .

STEP 9

For the teams with $104$ members, the squared difference is $(104 - 83.5)^2 = (20.5)^2 = 420.25$ .
Since there are $3$ such teams, we multiply by $3$ to get $3 \cdot 420.25 = 1260.75$ .

STEP 10

Adding those weighted squared differences, we get $2790.75 + 225 + 1260.75 = 4276.5$ .
To find the variance, we divide this sum by the total number of teams, which is $10$ .
So, $\sigma^2 = \frac{4276.5}{10} = 427.65$ .

STEP 11

The standard deviation, $\sigma$ , is just the square root of the variance.
So, $\sigma = \sqrt{427.65} \approx 20.68$ .
Rounding to the nearest hundredth gives us $\sigma \approx 20.68$ .

STEP 12

The standard deviation of the number of members in a randomly chosen team is approximately $20.68$ .

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Studdy Solution

STEP 1

What is this asking? If we randomly pick a track team, how spread out is the data of team sizes? Watch out! Don't forget to weigh the number of members by the number of teams with that many members.It's not just three data points!

STEP 2

1. Calculate the mean.2. Calculate the variance.3. Calculate the standard deviation.

STEP 3

First, let's **find the total number of teams**!We have 333 teams with 535353 members, 444 teams with 919191 members, and 333 teams with 104104104 members.So, we have a total of 3+4+3=103 + 4 + 3 = 103+4+3=10 teams!

STEP 4

STEP 5

To **calculate the mean**, μ\muμ, we'll divide the total number of runners by the total number of teams.So, μ=83510=83.5\mu = \frac{835}{10} = 83.5μ=10835​=83.5 runners per team on average.

STEP 6

The **variance**, σ2\sigma^2σ2, measures how spread out the data is.For each team size, we'll find the squared difference from the mean, multiply by the number of teams of that size, and then add those up.

STEP 7

For the teams with 535353 members, the squared difference is (53−83.5)2=(−30.5)2=930.25(53 - 83.5)^2 = (-30.5)^2 = 930.25(53−83.5)2=(−30.5)2=930.25.Since there are 333 such teams, we multiply by 333 to get 3⋅930.25=2790.753 \cdot 930.25 = 2790.753⋅930.25=2790.75.

STEP 8

For the teams with 919191 members, the squared difference is (91−83.5)2=(7.5)2=56.25(91 - 83.5)^2 = (7.5)^2 = 56.25(91−83.5)2=(7.5)2=56.25.Since there are 444 such teams, we multiply by 444 to get 4⋅56.25=2254 \cdot 56.25 = 2254⋅56.25=225.

STEP 9

For the teams with 104104104 members, the squared difference is (104−83.5)2=(20.5)2=420.25(104 - 83.5)^2 = (20.5)^2 = 420.25(104−83.5)2=(20.5)2=420.25.Since there are 333 such teams, we multiply by 333 to get 3⋅420.25=1260.753 \cdot 420.25 = 1260.753⋅420.25=1260.75.

STEP 10

STEP 11

The **standard deviation**, σ\sigmaσ, is just the square root of the variance.So, σ=427.65≈20.68\sigma = \sqrt{427.65} \approx 20.68σ=427.65​≈20.68.Rounding to the nearest hundredth gives us σ≈20.68\sigma \approx 20.68σ≈20.68.

STEP 12

The standard deviation of the number of members in a randomly chosen team is approximately 20.6820.6820.68.

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? If we randomly pick a track team, how spread out is the data of team sizes? Watch out! Don't forget to weigh the number of members by the number of teams with that many members.
It's not just three data points!

1. Calculate the mean.
2. Calculate the variance.
3. Calculate the standard deviation.

First, let's find the total number of teams!
We have $3$ teams with $53$ members, $4$ teams with $91$ members, and $3$ teams with $104$ members.
So, we have a total of $3 + 4 + 3 = 10$ teams!

To calculate the mean, $\mu$ , we'll divide the total number of runners by the total number of teams.
So, $\mu = \frac{835}{10} = 83.5$ runners per team on average.

The variance, $\sigma^2$ , measures how spread out the data is.
For each team size, we'll find the squared difference from the mean, multiply by the number of teams of that size, and then add those up.

For the teams with $53$ members, the squared difference is $(53 - 83.5)^2 = (-30.5)^2 = 930.25$ .
Since there are $3$ such teams, we multiply by $3$ to get $3 \cdot 930.25 = 2790.75$ .

For the teams with $91$ members, the squared difference is $(91 - 83.5)^2 = (7.5)^2 = 56.25$ .
Since there are $4$ such teams, we multiply by $4$ to get $4 \cdot 56.25 = 225$ .

For the teams with $104$ members, the squared difference is $(104 - 83.5)^2 = (20.5)^2 = 420.25$ .
Since there are $3$ such teams, we multiply by $3$ to get $3 \cdot 420.25 = 1260.75$ .

The standard deviation, $\sigma$ , is just the square root of the variance.
So, $\sigma = \sqrt{427.65} \approx 20.68$ .
Rounding to the nearest hundredth gives us $\sigma \approx 20.68$ .

The standard deviation of the number of members in a randomly chosen team is approximately $20.68$ .