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.
.45 is .05 away from the mean or .05/.023 = 2.17 standard deviations away from the mean. Your stats book should have a cumulative normal distribution table. Look up 2.17. I get .9850, meaning that the probability of observing a sample where more than 45% is 1-.985 = .015=1.5%
.35 is also 2.17 standard deviations from the mean. Take it from here.
so for c. do i subtract the amounts of the two
To find the probabilities of the given events using the density curve, we can calculate the area under the curve within the specified range. We will use a standard normal distribution for this calculation.
a. To find the probability that at least 45% of the sample believes violence is the most serious problem, we want to find the area to the right of 0.45 in the density curve. Let's denote this as P(X ≥ 0.45).
Using the standard normal distribution table or a statistical calculator, we can find the corresponding z-score for 0.45. The formula for the z-score is:
z = (x - μ) / σ
where x is the value we want to find the z-score for, μ is the mean, and σ is the standard deviation.
In this case, x = 0.45, μ = 0.4, and σ = 0.023. Plugging these values into the formula, we get:
z = (0.45 - 0.4) / 0.023 = 2.17 (approximately)
Looking up the area to the right of 2.17 in the standard normal distribution table or using a calculator, we find that the probability is approximately 0.015 (or 1.5%).
Therefore, P(X ≥ 0.45) ≈ 0.015.
b. To find the probability that less than 35% of the sample believes violence is the most serious problem, we want to find the area to the left of 0.35 in the density curve. Let's denote this as P(X < 0.35).
Using the same method as above, we calculate the z-score:
z = (0.35 - 0.4) / 0.023 = -2.17 (approximately)
Looking up the area to the left of -2.17 in the standard normal distribution table or using a calculator, we find that the probability is approximately 0.015 (or 1.5%).
However, we want to find the probability that the proportion is less than 0.35, so we need to subtract this probability from 1:
P(X < 0.35) = 1 - 0.015 ≈ 0.985
Therefore, P(X < 0.35) ≈ 0.985.
c. To find the probability that the sample proportion is between 0.35 and 0.45, we want to find the area between these two values in the density curve. Let's denote this as P(0.35 < X < 0.45).
Using the z-scores calculated previously, we can find the area to the right of -2.17 (P(X > -2.17)) and the area to the right of 2.17 (P(X > 2.17)):
P(X > -2.17) ≈ 0.985
P(X > 2.17) ≈ 0.015
To find the probability between 0.35 and 0.45, we subtract these two probabilities:
P(0.35 < X < 0.45) = P(X > -2.17) - P(X > 2.17) ≈ 0.985 - 0.015 ≈ 0.97
Therefore, P(0.35 < X < 0.45) ≈ 0.97.
To find the probabilities of these events using the normal density curve, we need to calculate the z-scores for the given proportions and then look up the corresponding probabilities in the standard normal distribution table.
a. The event is "at least 45% of the sample believes that violence is the school's most serious problem." In other words, we need to find the probability that the sample proportion (p) is greater than or equal to 0.45.
To calculate the z-score, we use the formula:
z = (x - μ) / σ
where x is the given proportion, μ is the mean, and σ is the standard deviation.
In this case, x = 0.45, μ = 0.4, and σ = 0.023.
z = (0.45 - 0.4) / 0.023 = 2.174
Now, we can look up the probability associated with the z-score of 2.174 in the standard normal distribution table. The probability of interest is the area to the right of the z-score.
Looking up the z-score in the table, we find that the probability is approximately 0.0147.
Therefore, the probability that at least 45% of the sample believes that violence is the school's most serious problem is approximately 0.0147.
b. The event is "less than 35% of the sample believes that violence is the most serious problem." In other words, we need to find the probability that the sample proportion (p) is less than 0.35.
Using the same steps as above, we calculate the z-score:
z = (0.35 - 0.4) / 0.023 = -2.174
We want to find the probability to the left of the z-score -2.174 in the standard normal distribution table.
Looking up the z-score in the table, we find the probability is approximately 0.0147.
However, since we want the probability that is less than 0.35, we need to subtract this probability from 1 (since the total area under the curve is 1).
1 - 0.0147 = 0.9853
Therefore, the probability that less than 35% of the sample believes that violence is the most serious problem is approximately 0.9853.
c. The event is "the sample proportion is between 0.35 and 0.45."
We need to find the probability that the sample proportion (p) lies between 0.35 and 0.45. To calculate this probability, we find the z-scores for each value and then subtract the corresponding probabilities.
For p = 0.35:
z1 = (0.35 - 0.4) / 0.023 = -2.174
For p = 0.45:
z2 = (0.45 - 0.4) / 0.023 = 2.174
Next, we find the probabilities associated with these z-scores using the standard normal distribution table:
Lower probability (z1) = approximately 0.0147
Upper probability (z2) = approximately 0.9853
To find the probability between these two values, we subtract the lower probability from the upper probability:
0.9853 - 0.0147 = 0.9706
Therefore, the probability that the sample proportion is between 0.35 and 0.45 is approximately 0.9706.
Remember that these probabilities are based on the assumption that the underlying distribution follows a normal curve with a mean of 0.4 and a standard deviation of 0.023.