Math

QuestionFind the modal class of wallaby heights and estimate the mean and standard deviation from the given frequency data.

Studdy Solution

STEP 1

Assumptions1. The table represents the heights of50 wallabies distributed in different height classes. . The modal class is the height class with the highest frequency.
3. The mean and standard deviation are to be estimated using the midpoint of each class as the representative height for that class.
4. The formula for mean is μ=(fx)\mu = \frac{\sum (f \cdot x)}{}, where ff is the frequency and xx is the midpoint of each class, and is the total number of observations.
5. The formula for standard deviation is $\sigma = \sqrt{\frac{\sum (f \cdot (x - \mu)^)}{}}$, where $f$ is the frequency, $x$ is the midpoint of each class, $\mu$ is the mean, and is the total number of observations.

STEP 2

To find the modal class, we look for the class with the highest frequency.
The modal class is the one with the highest frequency, which is 180x<190180 \leq x<190.

STEP 3

To estimate the mean, we first need to calculate the midpoint of each class. The midpoint is calculated as the average of the lower and upper limits of each class.
For example, the midpoint of the first class 150x<160150 \leq x<160 is calculated asMidpoint=150+1602Midpoint = \frac{150 +160}{2}

STEP 4

Calculate the midpoints for all classes in the same way.

STEP 5

Next, multiply each midpoint by its corresponding frequency to get fxf \cdot x for each class.

STEP 6

Add up all the fxf \cdot x values to get (fx)\sum (f \cdot x).

STEP 7

Divide (fx)\sum (f \cdot x) by the total number of wallabies (50) to estimate the mean height.
μ=(fx)\mu = \frac{\sum (f \cdot x)}{}

STEP 8

To estimate the standard deviation, we first need to calculate (xμ)2(x - \mu)^2 for each class.
For example, for the first class, this is calculated as(xμ)2=(MidpointMean)2(x - \mu)^2 = (Midpoint - Mean)^2

STEP 9

Calculate (xμ)2(x - \mu)^2 for all classes in the same way.

STEP 10

Next, multiply each (xμ)2(x - \mu)^2 by its corresponding frequency to get f(xμ)2f \cdot (x - \mu)^2 for each class.

STEP 11

Add up all the f \cdot (x - \mu)^ values to get (f(xμ))\sum (f \cdot (x - \mu)^).

STEP 12

Divide (f(xμ)2)\sum (f \cdot (x - \mu)^2) by the total number of wallabies (50) and then take the square root to estimate the standard deviation.
σ=(f(xμ)2)\sigma = \sqrt{\frac{\sum (f \cdot (x - \mu)^2)}{}}

STEP 13

Comment on the results. The mean gives us an estimate of the average height of the wallabies, while the standard deviation gives us an estimate of how much the heights vary around this average.

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