Question

several archeology students are unprepared for a surprise true/false test with 22 questions, and all of their answers are guessed. find the mean for the number of correct answers for such students. (round answer to one decimal place.) μ = find the standard deviation for the number of correct answers for such students. (round answer to two decimal places.) σ = give the range for the usual number of correct answers. (enter answer as an interval using square - brackets only with whole numbers.) usual values = would it be unusual for a student to pass by guessing and getting at least 14 correct answers? no, it would not be unusual yes, it would be unusual

Explanation:

Step1: Identify distribution type

This is a binomial distribution problem, where $n = 22$ (number of questions) and $p=0.5$ (probability of getting a true - false question correct by guessing).

Step2: Calculate the mean

The mean of a binomial distribution is given by $\mu=np$.
$\mu = 22\times0.5=11.0$

Step3: Calculate the standard deviation

The standard deviation of a binomial distribution is $\sigma=\sqrt{np(1 - p)}$.
$\sigma=\sqrt{22\times0.5\times(1 - 0.5)}=\sqrt{22\times0.5\times0.5}=\sqrt{5.5}\approx2.35$

Step4: Find the range for usual values

Usual values are within $2$ standard deviations of the mean. Lower limit: $\mu - 2\sigma=11-2\times2.35 = 11 - 4.7 = 6.3\approx6$. Upper limit: $\mu + 2\sigma=11 + 2\times2.35=11 + 4.7 = 15.7\approx16$. So the interval is $[6,16]$.

Step5: Determine if 14 is unusual

Since $14$ is within the interval $[6,16]$, it is not unusual.

Answer:

$\mu = 11.0$
$\sigma\approx2.35$
usual values = $[6,16]$
No, it would not be unusual