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:

Question

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.

Answer 1

So, you are assuming a true population with a normal distribution of 0.4; 0.023)

A)
Convert 0.45 to a Z score: (0.45 - 0.40) / 0.023 = +2.17
Look up on a z-table chart. Greater than +2.17 yields .0150 = 1.5%

B)
<= 0.35 has the exact opposite Z score: (0.35 - 0.40) / 0.023 = -2.17
Less than -2.17 yields .0150 = 1.5% (same answer as A)

Answer 2

Forgot C. That's probably obvious after you have A + B:

C)
100% - 1.5% - 1.5% = 97%

Answer 3

To find the probabilities of the given events, you can use the normal distribution with mean 0.4 and standard deviation 0.023.

a. To find the probability that at least 45% of the sample believes violence is the school's most serious problem, you need to find the area under the density curve to the right of 45%.

Since the normal distribution is symmetrical, you can also find the probability to the left of 55% and subtract it from 1.

Using a Z-table or a calculator, you can find the z-score corresponding to 45% as follows:

z = (x - µ) / σ
z = (0.45 - 0.4) / 0.023
z = 2.1739

Looking up the area to the right of 2.1739 in the Z-table, the probability is 0.0149. Subtracting this from 1 gives the probability of at least 45% believing violence is the most serious problem: 1 - 0.0149 = 0.9851, or 98.51%.

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

Using the same method as in part a, you can find the z-score corresponding to 35%:

z = (x - µ) / σ
z = (0.35 - 0.4) / 0.023
z = -2.1739

Looking up the area to the left of -2.1739 in the Z-table, the probability is 0.0149. Therefore, the probability of less than 35% believing violence is the most serious problem is 0.0149, or 1.49%.

c. To find the probability that the sample proportion is between 0.35 and 0.45, you need to find the area under the density curve between these two values.

Using the z-scores found in parts a and b, you can find the area to the right of 35% and subtract the area to the right of 45%. However, since both these probabilities are already known from parts a and b, you can simply subtract them: 0.9851 - 0.0149 = 0.9702, or 97.02%.

Therefore, the probability that the sample proportion is between 0.35 and 0.45 is 0.9702, or 97.02%.