Question

assume that a procedure yields a binomial distribution with n = 28 trials and the probability of success for one trial is p = 0.42.
find the mean for this binomial distribution.
(round answer to one decimal place.)
μ =
find the standard deviation for this distribution.
(round answer to two decimal places.)
σ =
will the probability histogram for the binomial random variable be approximately bell shaped?

Explanation:

Step1: Recall binomial mean formula

The mean \(\mu\) of a binomial distribution is given by \(\mu = np\), where \(n\) is the number of trials and \(p\) is the probability of success on a single trial. Here, \(n = 28\) and \(p = 0.42\).

$$ \mu = 28\times0.42 $$

Step2: Calculate the mean

$$ 28\times0.42 = 11.76 $$

Step3: Round to one decimal place

Rounding \(11.76\) to one decimal place gives \(11.8\).

Step4: Recall binomial standard deviation formula

The standard deviation \(\sigma\) of a binomial distribution is given by \(\sigma=\sqrt{np(1 - p)}\). We know \(n = 28\), \(p = 0.42\), so \(1-p=1 - 0.42 = 0.58\).

$$ \sigma=\sqrt{28\times0.42\times0.58} $$

Step5: Calculate the value inside the square root

First, calculate \(28\times0.42\times0.58\):

$$ 28\times0.42 = 11.76; \quad 11.76\times0.58 = 6.8208 $$

Step6: Take the square root and round

Now, take the square root of \(6.8208\): \(\sqrt{6.8208}\approx2.61\) (rounded to two decimal places).

Answer:

For the mean \(\mu\): \(11.8\)

For the standard deviation \(\sigma\): \(2.61\)