4. The proportion of the students in private schools is around 11% with the standard deviation of 2%. A ramdom sample of 450 students from a wide geographic area indicated that 55 attended private schools. Estimate the true proportion of students attending private schools with 95% confidence.
Well, if we're talking about private schools, it sounds like we're dealing with some pretty exclusive education here. Let's crunch some numbers and see what we come up with!
Given that the sample size is 450 and 55 students attended private schools, we can calculate the sample proportion by dividing the number of students attending private schools by the total sample size:
Sample proportion = 55 / 450 = 0.1222 (rounded to four decimal places)
Now, let's calculate the margin of error, which represents a range around the sample proportion within which we're 95% confident the true proportion exists. The formula for the margin of error is z * (sqrt(p * (1-p) / n)), where z represents the z-score for a given confidence level, p is the sample proportion, and n is the sample size. With a 95% confidence level, the z-score is approximately 1.96.
Margin of error = 1.96 * (sqrt(0.1222 * (1 - 0.1222) / 450)) ≈ 0.0257 (rounded to four decimal places)
Finally, to calculate the confidence interval, we subtract and add the margin of error from the sample proportion:
Confidence interval = Sample proportion ± Margin of error
Confidence interval = 0.1222 ± 0.0257
Confidence interval ≈ [0.0965, 0.1479] (rounded to four decimal places)
Therefore, with 95% confidence, we estimate that the true proportion of students attending private schools in the wide geographic area lies within the range of approximately 9.65% to 14.79%.
Just remember, these results are based on a sample, so take them with a grain of salt... or maybe with a clown nose.
To estimate the true proportion of students attending private schools with 95% confidence, we can use the formula for a confidence interval for proportions.
Step 1: Calculate the sample proportion
In our case, the sample proportion of students attending private schools is 55 out of 450, which is 55/450 = 0.1222.
Step 2: Calculate the standard error
The standard error (SE) for proportions is calculated as:
SE = sqrt((p*(1-p))/n)
Where p is the sample proportion (0.1222) and n is the sample size (450).
SE = sqrt((0.1222*(1-0.1222))/450) ≈ 0.015
Step 3: Determine the critical value
For a 95% confidence interval, the critical value can be obtained from the z-table or using a calculator. For this example, the critical value is approximately 1.96.
Step 4: Calculate the margin of error
The margin of error (ME) is calculated by multiplying the critical value by the standard error.
ME = 1.96 * 0.015 ≈ 0.0294
Step 5: Calculate the confidence interval
The confidence interval is calculated as the sample proportion plus/minus the margin of error.
CI = sample proportion ± margin of error
CI = 0.1222 ± 0.0294
Step 6: Interpret the confidence interval
The confidence interval gives us a range within which we can be 95% confident that the true proportion of students attending private schools lies. In this case, the confidence interval is (0.0928, 0.1516), which means we can be 95% confident that the true proportion of students attending private schools is between 9.28% and 15.16%.
Therefore, the estimate of the true proportion of students attending private schools with 95% confidence is between 9.28% and 15.16%.
To estimate the true proportion of students attending private schools with 95% confidence, we can use the formula for estimating a population proportion.
The formula is:
Confidence Interval = Sample Proportion ± (Z * Standard Error)
Where:
- Sample Proportion is the number of students attending private schools in the sample divided by the total sample size.
- Z is the z-score corresponding to the desired confidence level. In this case, since we want a 95% confidence level, the corresponding z-score is 1.96.
- Standard Error is the standard deviation of the population proportion.
First, let's calculate the sample proportion:
Sample Proportion = Number of students attending private schools / Total sample size
= 55 / 450
≈ 0.1222
Next, we need to calculate the standard error:
Standard Error = √[(Sample Proportion * (1 - Sample Proportion)) / Sample Size]
= √[(0.1222 * (1 - 0.1222)) / 450]
≈ √[(0.1222 * 0.8778) / 450]
≈ √(0.1072 / 450)
≈ √0.0002382
≈ 0.0154
Now we can calculate the confidence interval:
Confidence Interval = Sample Proportion ± (Z * Standard Error)
= 0.1222 ± (1.96 * 0.0154)
= 0.1222 ± 0.0302
Finally, we can interpret the confidence interval. The lower bound of the confidence interval is 0.1222 - 0.0302 ≈ 0.092, and the upper bound is 0.1222 + 0.0302 ≈ 0.152. Therefore, we can estimate with 95% confidence that the true proportion of students attending private schools is between approximately 9.2% and 15.2%.