A random sample of size n = 270 is taken from a population with a population proportion p = 0.58. [You may find it useful to reference the z table.]
a. Calculate the expected value and the standard error for the sampling distribution of the sample proportion. (Round "expected value" to 2 decimal places and "standard error" to 4 decimal places.)
b. What is the probability that the sample proportion is between 0.50 and 0.60? (Round final answer to 4 decimal places.)
c. What is the probability that the sample proportion is less than 0.50? (Round final answer to 4 decimal places.)
a. The expected value of the sample proportion is equal to the population proportion, so expected value = p = 0.58.
The standard error is calculated using the formula:
standard error = sqrt((p(1-p))/n)
Substituting the values in, we get:
standard error = sqrt((0.58(1-0.58))/270) ≈ 0.0336
b. To find the probability that the sample proportion is between 0.50 and 0.60, we need to calculate the z-scores for these values and then use the z-table to find the respective probabilities.
The z-score for 0.50 is calculated using the formula:
z = (x - μ) / σ
Substituting the values in, we get:
z1 = (0.50 - 0.58) / 0.0336 ≈ -2.38
Similarly, the z-score for 0.60 is:
z2 = (0.60 - 0.58) / 0.0336 ≈ 0.60
Using the z-table, we find the corresponding probabilities:
P(0.50 < x < 0.60) ≈ P(-2.38 < z < 0.60) ≈ P(z < 0.60) - P(z < -2.38)
Using the z-table, we find the respective probabilities:
P(z < 0.60) ≈ 0.7257
P(z < -2.38) ≈ 0.0088
P(0.50 < x < 0.60) ≈ 0.7257 - 0.0088 = 0.7169
So, the probability that the sample proportion is between 0.50 and 0.60 is approximately 0.7169.
c. To find the probability that the sample proportion is less than 0.50, we use the z-score formula and the z-table.
The z-score for 0.50 is:
z = (x - μ) / σ
Substituting the values in, we get:
z = (0.50 - 0.58) / 0.0336 ≈ -2.38
Using the z-table, we find the probability:
P(x < 0.50) ≈ P(z < -2.38) ≈ 0.0088
So, the probability that the sample proportion is less than 0.50 is approximately 0.0088.
To calculate the expected value and the standard error for the sampling distribution of the sample proportion, we can use the formulas:
Expected value (μ) = p = 0.58
Standard error (SE) = √[(p * (1 - p)) / n]
Where:
- p is the population proportion (given as 0.58)
- n is the sample size (given as 270)
a. Calculating the expected value:
μ = 0.58
b. Calculating the standard error:
SE = √[(0.58 * (1 - 0.58)) / 270]
= √[(0.58 * 0.42) / 270]
= √[0.2436 / 270]
≈ √0.0009022
≈ 0.03004 (rounded to 4 decimal places)
c. To calculate the probability that the sample proportion is between 0.50 and 0.60, we need to find the z-scores for both values and then find the area between those two z-scores using the z-table.
For 0.50:
z = (0.50 - p) / SE
= (0.50 - 0.58) / 0.03004
= -0.08 / 0.03004
≈ -2.6648
For 0.60:
z = (0.60 - p) / SE
= (0.60 - 0.58) / 0.03004
= 0.02 / 0.03004
≈ 0.6662
Using the z-table, we can look up the area to the left of the z-scores -2.6648 and 0.6662. We then subtract the smaller area from the larger area to find the probability between those two values.
P(0.50 ≤ p ≤ 0.60) = P(z ≤ 0.6662) - P(z ≤ -2.6648)
Looking up these values in the z-table, we find:
P(z ≤ 0.6662) ≈ 0.7469
P(z ≤ -2.6648) ≈ 0.0038
P(0.50 ≤ p ≤ 0.60) = 0.7469 - 0.0038
≈ 0.7431 (rounded to 4 decimal places)
Therefore, the probability that the sample proportion is between 0.50 and 0.60 is approximately 0.7431.
d. To calculate the probability that the sample proportion is less than 0.50, we need to find the z-score for 0.50 and then find the area to the left of that z-score using the z-table.
Using the same calculation as in part c for 0.50, we found the z-score to be approximately -2.6648.
P(p < 0.50) = P(z < -2.6648)
Using the z-table, we find:
P(z < -2.6648) ≈ 0.0038
Therefore, the probability that the sample proportion is less than 0.50 is approximately 0.0038.
To calculate the expected value and standard error for the sampling distribution of the sample proportion, we will use the following formulas:
a. The expected value (mean) of the sampling distribution of the sample proportion (p̂) is equal to the population proportion (p). So, in this case, the expected value is 0.58.
b. The standard error (SE) of the sampling distribution of the sample proportion can be calculated using the formula:
SE = sqrt((p*(1-p))/n)
Substituting the given values:
SE = sqrt((0.58*(1-0.58))/270)
SE ≈ 0.0269 (rounded to 4 decimal places)
c. To find the probability that the sample proportion is between 0.50 and 0.60, we need to calculate the z-scores for these values and then use the z-table.
A z-score (z) can be found using the formula:
z = (x - μ) / SE
Where x is the value of interest, μ is the expected value, and SE is the standard error.
For 0.50:
z1 = (0.50 - 0.58) / 0.0269
For 0.60:
z2 = (0.60 - 0.58) / 0.0269
Next, we look up the z-scores in the z-table to find the corresponding probabilities. Subtracting the probability for z1 from the probability for z2 will give us the desired probability.
c. To find the probability that the sample proportion is less than 0.50, we need to calculate the z-score for this value and then use the z-table to find the corresponding probability.
For 0.50:
z = (0.50 - 0.58) / 0.0269
Then, we look up the z-score in the z-table to find the probability that the sample proportion is less than 0.50.