Violence in School, I An SRS of 400 American adults is asked “What do you think is the most serious problem facing our schools?” Suppose that in fact 40% of all adults would answer “violence” if asked this question. The proportion p of the sample who answers “violence” will vary in repeated sampling. In fact, we can assign probabilities to values of p using the normal density curve with mean 0.4 and standard deviation 0.023. Use the density curve to find the probabilities of the following event:

a.At least 45% of the sample believes that violence is the school’s most serious problem.

b.Less than 35% of the sample believes that violence is the most serious problem.

c.The sample proportion is between 0.35 and 0.45.

To find the probabilities of these events using the density curve, we need to calculate the area under the curve corresponding to each event.

a. To find the probability that at least 45% of the sample believes violence is the most serious problem, we need to find the area to the right of the value 0.45 on the density curve. Using a standard normal distribution, we can convert this value to a z-score using the formula:

z = (x - μ)/σ

where x is the value we want to find the area to the right of (0.45), μ is the mean (0.4), and σ is the standard deviation (0.023).

Calculating the z-score:

z = (0.45 - 0.4)/0.023
z = 2.1739

Using a z-table or a statistical calculator, we can find the area to the right of the z-score 2.1739. This area represents the probability of at least 45% of the sample believing violence is the most serious problem.

b. To find the probability that less than 35% of the sample believes violence is the most serious problem, we need to find the area to the left of the value 0.35 on the density curve.

Calculating the z-score:

z = (0.35 - 0.4)/0.023
z = -2.1739

Using a z-table or a statistical calculator, we can find the area to the left of the z-score -2.1739. This area represents the probability of less than 35% of the sample believing violence is the most serious problem.

c. To find the probability that the sample proportion is between 0.35 and 0.45, we need to find the area between the values 0.35 and 0.45 on the density curve.

Calculating the z-scores:

z1 = (0.35 - 0.4)/0.023
z1 = -2.1739

z2 = (0.45 - 0.4)/0.023
z2 = 2.1739

Using a z-table or a statistical calculator, we can find the area to the left of the z-score -2.1739 and the area to the right of the z-score 2.1739. Then, we subtract the area of the left side from the area of the right side to find the area between the two z-scores. This area represents the probability of the sample proportion being between 0.35 and 0.45.