The Democrat and Chronicle reported that 25% of the flights arriving at the San Diego airport during the first five months of 2001 were late (Democrat and Chronicle, July 23, 2001). Assume the population proportion is p = .25.
1. Calculate () with a sample size of 1,200 flights (to 4 decimals).
2. What is the probability that the sample proportion will be within +/- .03 of the population proportion if a sample of size 1,200 is selected (to 4 decimals)?
3. What is the probability that the sample proportion will be within +/- .03 of the population proportion if a sample of size 600 is selected (to 4 decimals)?
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1) (sqrt(.25*(1-.25))/1200) = .0125
2) .03/.0125 = 2.4 (z-score)
-.03/.0125 = -2.4 (z-score)
Find value using standard normal distribution table:
2.4 = .9918
-2.4 = .0082
.9918-.0082 = .9836
3) (sqrt(.25*(1-.25))/600) = .0177
.03/.0177 = 1.69 (z-score)
-.03/.0177 = -1.69 (z-score)
Find value using standard normal distribution table:
1.69 = .9545
-1.69 - .0455
.9545 - .0455 = .9090
To calculate these probabilities, we will use the concept of sampling distribution and assume that the sample proportion follows a normal distribution. The mean of the sampling distribution of the sample proportion is equal to the population proportion p, and the standard deviation is given by the formula sqrt((p*(1-p))/n), where n is the sample size.
1. Calculate the standard deviation ()
Given:
Population proportion, p = 0.25
Sample size, n = 1,200
Using the formula sqrt((p*(1-p))/n), we can calculate the standard deviation:
= sqrt((0.25*(1-0.25))/1200)
= sqrt(0.25*0.75/1200)
= sqrt(0.1875/1200)
= sqrt(0.00015625)
≈ 0.0125 (rounded to 4 decimals)
2. Calculate the probability that the sample proportion will be within +/- .03 of the population proportion if a sample of size 1,200 is selected.
Given:
Population proportion, p = 0.25
Sample size, n = 1,200
Margin of error, E = 0.03
To calculate the probability, we need to find the z-scores corresponding to the sample proportion within the given margin of error. The z-score can be calculated using the formula:
z = (P - p) / ()
Where P is the sample proportion.
Since we want to find the probability for both sides of the margin of error, we need to find two z-scores.
For the upper z-score:
z_upper = (p + E - p) / ()
= E / ()
= 0.03 / 0.0125
For the lower z-score:
z_lower = (p - E - p) / ()
= -E / ()
= -0.03 / 0.0125
Using a standard normal distribution table or a calculator, we can find the probabilities associated with these z-scores.
P(z < z_upper) ≈ P(z < 2.4) (rounded to 1 decimal)
P(z > z_lower) ≈ P(z > -2.4) (rounded to 1 decimal)
The probability between the two z-scores is:
P(z_lower < z < z_upper) = P(z > z_lower) - P(z > z_upper)
Therefore, the probability that the sample proportion will be within +/- .03 of the population proportion for a sample size of 1,200 is approximately:
P(z_lower < z < z_upper) ≈ P(z > -2.4) - P(z > 2.4)
3. Calculate the probability that the sample proportion will be within +/- .03 of the population proportion if a sample of size 600 is selected.
Following the same steps as in question 2, you will calculate the z-scores for a sample size of 600:
z_upper = 0.03 / ()
z_lower = -0.03 / ()
Then find the corresponding probabilities using a standard normal distribution table or calculator.
Note: The standard normal distribution table provides the probabilities for the z-score to the left (P(z < z)) or to the right (P(z > z)). To calculate the probability between two z-scores, you need to subtract the probability of the larger z-score from the probability of the smaller z-score.