Question

suppose that the distribution for total amounts spent by students vacationing for a week in florida is normally distributed with a mean of 650 and a standard deviation of 120 . suppose you take a simple random sample (srs) of 10 students from this distribution. what is the probability that a srs of 10 students will spend an average of between 600 and 700 dollars? round to five decimal places.

Answer 1

The probability that the SRS of 10 students will spend an average of between 600 and 700 dollars is 0.8132.

Let x be the total amount spent by students.

x follows normal distribution with mean μ = 650,

                                  standard deviation δ = 120

We take a simple random sample of size n = 10

We are asked to find average spending of 10 students ( $x$ ) is between 600 and 700

We have to find P( 600 <= $x$ <= 700)

According to the sampling distribution of the sample mean $x$, it follows an approximately normal distribution with mean μ{$x$} = μ and standard deviation  δ{$x$} = {δ}/{√n}

Therefore here mean of $x$,( μ{$x$} ) = 650 and

standard deviation of $x$, δ{$x$} = {120}/{√10} = 37.9473

The probability that the SRS of 10 students will spend an average of between 600 and 700 dollars is,

Let Z= x - μ / δ

Z₁ = 600 - 650 / 37.9473 = -1.32 similarly

Z₂ = 700 - 650 / 37.9473 = 1.32

From standard normal distribution table, P( -1.32 <  $x$  <  1.32) = 0.8132

The probability that the SRS of 10 students will spend an average of between 600 and 700 dollars is 0.8132

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