In a population of 10,000 adults, 20% are smokers. A simple random sample of 600 of the adults is drawn.

a)The SE of the number of smokers in the sample is closest to

9.45
9.5
9.55
9.6
9.65
9.7
9.75
9.8
9.85
9.9

b)The chance that there are fewer than 115 smokers in the sample is closest to (pick the best of the options below, even if you can calculate the probability more accurately)

10%
20%
30%
40%
50%
60%
70%
80%
90%

SE=9.5

chance=30%

and the Expected smoker #?

120

thanks

Can you give the steps that led to these answers?

To find the standard error (SE) of the number of smokers in the sample, we can use the formula:

SE = √(p(1-p)/n)

Where p is the proportion of smokers in the population and n is the sample size.

In this case, the proportion of smokers in the population is given as 20% or 0.20, and the sample size is 600.

a) To find the SE, we can substitute these values into the formula:

SE = √(0.20(1-0.20)/600)
SE = √(0.16/600)
SE ≈ √0.00026
SE ≈ 0.016

Rounding this to two decimal places, the closest answer in the given options is 9.6. Therefore, the SE of the number of smokers in the sample is closest to 9.6.

b) To find the probability of there being fewer than 115 smokers in the sample, we can use the normal distribution approximation.

First, we need to find the mean of the number of smokers in the sample. The mean can be calculated as:

mean = n * p
mean = 600 * 0.20
mean = 120

Next, we need to find the standard deviation (SD) of the number of smokers in the sample. The SD can be calculated as:

SD = √(n * p * (1-p))
SD = √(600 * 0.20 * (1-0.20))
SD = √(120 * 0.80)
SD ≈ √96
SD ≈ 9.8

Using the normal distribution table or a statistical calculator, we can find the probability of there being fewer than 115 smokers in the sample. Since the mean is 120 and the standard deviation is 9.8, we can calculate the z-score as:

z = (x - mean) / SD
z = (115 - 120) / 9.8
z ≈ -0.51

Using the z-score, we can find the corresponding probability. Looking it up in the normal distribution table, we find that the probability is approximately 0.305.

Considering the given options, the closest answer is 30%. Therefore, the chance that there are fewer than 115 smokers in the sample is closest to 30%.