Question

an online spinner has 2 colored regions—blue and yellow. according to the website, the probability that the spinner lands in the blue region on any spin is 0.80. assume that this claim is correct. suppose we spin the spinner 12 times. let x = the number of times that the spinner lands in the blue region.
(a) calculate the mean of x.
round your answer to the nearest tenth.
interpret the mean of x.
if 12 spins are done with the spinner, we expect about spins will land in the blue region, on average.
(b) calculate the standard deviation of x.
round your answer to 2 decimal places.
interpret the standard deviation of x.
if 12 spins are done with the spinner, the number of spins that land in the blue region would typically vary from by about spins.

Explanation:

Step1: Identify the distribution

This is a binomial distribution problem where \( n = 12 \) (number of trials, spins) and \( p = 0.80 \) (probability of success, landing on blue). The mean of a binomial distribution is given by \( \mu = np \).
\( \mu = 12\times0.80 \)

Step2: Calculate the mean

\( 12\times0.80 = 9.6 \)

Step1: Recall the formula for standard deviation of binomial distribution

The standard deviation \( \sigma \) of a binomial distribution is given by \( \sigma=\sqrt{np(1 - p)} \), where \( n = 12 \), \( p = 0.80 \), and \( 1-p=0.20 \).

Step2: Substitute the values into the formula

\( \sigma=\sqrt{12\times0.80\times0.20} \)
First, calculate the product inside the square root: \( 12\times0.80\times0.20 = 12\times0.16 = 1.92 \)
Then, take the square root: \( \sigma=\sqrt{1.92}\approx1.39 \) (rounded to two decimal places)

The mean of a binomial random variable \( X \) (number of blue spins in 12 trials) being \( \mu = 9.6 \) means that if we were to spin the spinner 12 times many times, on average, we would expect about 9.6 of those spins to land in the blue region. Since we can't have a fraction of a spin, it implies that over many sets of 12 spins, the average number of blue spins approaches 9.6.

The standard deviation of \( X \) being approximately 1.39 means that the number of spins that land in the blue region when 12 spins are done will typically vary from the mean (9.6) by about 1.39 spins. In other words, most of the time, the number of blue spins in 12 trials will be within about 1.39 of 9.6 (e.g., between \( 9.6 - 1.39 = 8.21 \) and \( 9.6+1.39 = 10.99 \), so roughly 8 to 11 spins for practical purposes).

Answer:

9.6