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 the given events, we can use the normal distribution and the Z-score formula. The Z-score formula allows us to standardize values from a normal distribution and find probabilities associated with those values.
The formula for calculating the Z-score is:
Z = (x - μ) / σ
Where:
Z is the Z-score
x is the value we want to find the probability for
μ is the mean of the distribution
σ is the standard deviation of the distribution
a. To find the probability that at least 45% of the sample believes violence is the school's most serious problem, we need to calculate the area under the normal curve to the right of 45%. Since we're given the mean (μ = 0.4) and the standard deviation (σ = 0.023), we can calculate the Z-score for 45% as follows:
Z = (0.45 - 0.4) / 0.023 = 2.17
Using a Z-table or a statistics calculator, we can find that the probability of getting a Z-score of 2.17 or greater is approximately 0.0157. Therefore, the probability that at least 45% of the sample believes violence is the school's most serious problem is 0.0157.
b. To find the probability that less than 35% of the sample believes violence is the most serious problem, we need to calculate the area under the normal curve to the left of 35%. Using the same formula and values as in part a, we can calculate the Z-score for 35% as follows:
Z = (0.35 - 0.4) / 0.023 = -2.17
Using a Z-table or a statistics calculator, we can find that the probability of getting a Z-score of -2.17 or less is approximately 0.0157. However, since we're interested in the probability of getting less than 35%, we need to calculate the area to the left of -2.17 and subtract it from 1. Therefore, the probability that less than 35% of the sample believes violence is the most serious problem is 1 - 0.0157 = 0.9843.
c. To find the probability that the sample proportion is between 0.35 and 0.45, we need to calculate the area under the normal curve between these two values. Similar to parts a and b, we can calculate the Z-scores for 0.35 and 0.45:
Z1 = (0.35 - 0.4) / 0.023 = -2.17
Z2 = (0.45 - 0.4) / 0.023 = 2.17
Using a Z-table or a statistics calculator, we can find the probabilities associated with these Z-scores. The probability of getting a Z-score less than -2.17 is approximately 0.0157, and the probability of getting a Z-score less than 2.17 is also approximately 0.0157. Therefore, the probability that the sample proportion is between 0.35 and 0.45 is 0.0157 + 0.0157 = 0.0314.
In summary:
a. The probability of at least 45% of the sample believing violence is the school's most serious problem is 0.0157.
b. The probability of less than 35% of the sample believing violence is the most serious problem is 0.9843.
c. The probability that the sample proportion is between 0.35 and 0.45 is 0.0314.