A sample of 2,000 people was asked how many cups of coffee they drink in the morning. Five hundred people said they do not drink coffee, eight hundred people said they drink one cup, six hundred said they drink two cups and one hundred said they drink three cups of coffee in the morning.
a. Determine the expected number of cups of coffee consumed each morning. (Round to two decimal places)
b. Determine the standard deviation for this sample. (Round to two decimal places)

In statistics and probability analysis, the expected value "is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values".

The variance of a random variable Var(X) is the expected value of the squared deviation from the mean of X, E(X).

And the standard deviation of a random variable X is just the square root of the variance.

Part a

The info given is:

N=2000 represent the sample

Number of people that said they do not drink coffee =500

Number of people that said they drink one cup =800

Number of people that said they drink two cups =600

Number of people that said they drink three cups of coffee =100

Let X represent the random variable who represent the number of cups of coffee consumed each morning

Now we can find the probability for each value of X

P(X=0)=500/2000=1/4, people that said they do not drink coffee

P(X=1)=800/2000=2/5, people that said they drink one cup

P(X=2)=600/2000=3/10, people that said they drink two cups

P(X=-3)=100/2000=1/20. people that said they drink three cups of coffee

So then the random variable is given by this table

X | 0 | 1 | 2 | 3 |

P(X) |1/4 | 2/5 | 3/10 | 1/20 |

In order to calculate the expected value we can use the following formula:

[tex]E(X)=\sum_{i=1}^n X_i P(X_i)[/tex]

And if we use the values obtained we got:

[tex]E(X)=(0)*(\frac{1}{4})+(1)(\frac{2}{5})+(2)(\frac{3}{10})+(3)(\frac{1}{20})=\frac{23}{20}=1.15[/tex]

Part b

In order to find the standard deviation we need to find first the second moment, given by :

[tex]E(X^2)=\sum_{i=1}^n X^2_i P(X_i)[/tex]

And using the formula we got:

[tex]E(X^2)=(0)*(\frac{1}{4})+(1)(\frac{2}{5})+(4)(\frac{3}{10})+(9)(\frac{1}{20})=\frac{41}{20}=2.05[/tex]

Then we can find the variance with the following formula:

[tex]Var(X)=E(X^2)-[E(X)]^2 =2.05-(1.15)^2 =0.7275[/tex]

And then the standard deviation would be given by:

[tex]Sd(X)=\sqrt{Var(X)}=\sqrt{0.7275}=0.8529[/tex]