Question

A poll agency reports that 75% of teenagers aged 12-17 own smartphones. A random sample of 234 teenagers is drawn. Round your answers to four decimal places as needed. Part 1. Find the mean . Part 2. out of 6 Find the standard deviation

Answer 1

The mean of the random sample is 175.5. The standard deviation is approximately 0.0283.

Step-by-step explanation:

Part 1:

To find the mean, we need to multiply the percentage of teenagers owning smartphones by the total number of teenagers in the random sample:

Mean = 75% * 234 = 0.75 * 234 = 175.5

Part 2:

To find the standard deviation, we can use the formula for standard deviation of a proportion:

Standard Deviation = sqrt[(p * (1 - p))/(n)], where p is the proportion and n is the sample size.

Standard Deviation = sqrt[(0.75 * (1 - 0.75))/(234)] = sqrt[(0.1875)/(234)] = sqrt[0.000801282] = 0.0283 (rounded to four decimal places).

Answer 2

If our random variable of interest for this case is X="the number of teenagers between 12-17 with smartphone" we can model the variable with this distribution:

`X \sim Binom(n=234, p=0.75)`

And the mean for this case would be:

`E(X) =np = 234*0.75= 175.5`

And the standard deviation would be given by:

`\sigma =√(np(1-p))= √(234*0.75*(1-0.75))= 6.624`

Step-by-step explanation:

If our random variable of interest for this case is X="the number of teenagers between 12-17 with smartphone" we can model the variable with this distribution:

`X \sim Binom(n=234, p=0.75)`

And the mean for this case would be:

`E(X) =np = 234*0.75= 175.5`

And the standard deviation would be given by:

`\sigma =√(np(1-p))= √(234*0.75*(1-0.75))= 6.624`