Question

A florist determines the probabilities for the number of flower arrangements they deliver each day.

x 19 20 21 22 23
P(x) 0.21 0.22 0.30 0.15 0.12

Mean = Find the mean, variance, and standard deviation of the distribution rounded to 4 decimal places.

Variance =

Standard Deviation =

Approximately how many arrangements should the florist expect to deliver each week, rounded to the nearest whole number?

Answer 1

Answer:

a) mean = 20.75

b) variance = 1.6274

c) standard deviation = 1.2757

d) 145 arrangements per week.

Step-by-step explanation:

We have the following distribution:

x       19      20    21     22      23

P(x) 0.21  0.22  0.30  0.15    0.12

Mean: Is the average of the data. To find the mean, you need to take every x and multiply it by its corresponding P(x) and sum these values. The result will be the mean.

μ =∑ (x·P(x))

Therefore, in this case we have:

μ = 19(0.21) + 20(0.22) + 21(0.30) + 22(0.15) + 23 (0.12) = 20.75

Thus, the mean is 20.75

Variance: To calculate the variance you need to subtract the mean to every value x, then square the result and multiply it by the probability. Finally, you sum up these results and you get the variance.

In this case we will have

19 - 20.75 = -1.75 ⇒ (-1.75)²(0.21) = 0.6431

20 - 20.75 = -.75 ⇒(0.75)²(0.22) = 0.1237

21 - 20.75 = 0.25 ⇒(0.25)²(0.30) = 0.0187

22 - 20.75 = 1.25 ⇒(1.25)²(0.15) = 0.2344

23 - 20.75 = 2.25 ⇒ (2.25)²(0.12) = 0.6075

Now we need to sum up these results:

0.6431 + 0.1237 + 0.0187 + 0.2344 + 0.6075 = 1.6274

The variance is 1.6274.

Standard Deviation: It is the root of the variance.

Therefore, the standard deviation will be √1.6274 = 1.2757

b) How many arrangements should the florist expect to deliver each week, rounded to the nearest whole number?

To answer this question we'll work with the mean, he expects to deliver 20.75 arrangements per day, therefore, per week he'd deliver:

20.75 (7) = 145.25 = 145 arrangements.

Answer 2

The mean of the given data is 20.75.

The variance of the given data is 1.62.

The standard deviation is 1.27.

There are 145 arrangements should the florist expect to deliver each week.

Given

A florist determines the probabilities for the number of flower arrangements they deliver each day.

x       19 20 21 22 23

P(x)  0.21 0.22 0.30 0.15 0.12

Mean;

Mean is the average of all the data.

$\rm \mu =\sum (x\times P(x))\\\\\mu == 19(0.21) + 20(0.22) + 21(0.30) + 22(0.15) + 23 (0.12) \\\\ \mu= 20.75$

The mean of the given data is 20.75.

Variance;

For the variance you need to subtract the mean to every value x, then square the result and multiply it by the probability.

$\rm 19 - 20.75 = -1.75 = (-1.75)^2(0.21) = 0.6431\\\\20 - 20.75 = -.75 =(0.75)^2(0.22) = 0.1237\\\\21 - 20.75 = 0.25 =(0.25)^2(0.30) = 0.0188\\\\22 - 20.75 = 1.25 =(1.25)^2(0.15) = 0.2344\\\\ 23 - 20.75 = 2.25 = (2.25)^2(0.12) = 0.6075$

Variance = 0.6431 + 0.1237 + 0.0187 + 0.2344 + 0.6075 = 1.6274

The variance of the given data is 1.62.

Standard Deviation;

Standard Deviation is defined as the is the root of the variance.

$\rm Standard \ deviation=\sqrt{variance} \\\\Standard \ deviation=\sqrt{1.42} \\\\ Standard \ deviation=1.27$

The standard deviation is 1.27.

2. How many arrangements should the florist expect to deliver each week?

$\rm = 20.75 \times 7 = 145.25 = 145\ arrangements.$

There are 145 arrangements should the florist expect to deliver each week.

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