suppose a sample of 2404 tenth graders is drawn.Of the sampled, 601 read at or below the eighth grade level. using the data, construct the 98% confidence interval for the population proportion of tenth graders reading at or below the eighth grade level.
To construct the confidence interval for the population proportion, we can use the formula:
Confidence interval = sample proportion ± margin of error
First, let's calculate the sample proportion:
Sample proportion = (Number of students reading at or below eighth grade level / Total number of students)
Sample proportion = 601 / 2404
Sample proportion = 0.249
Next, we need to calculate the margin of error. The margin of error can be calculated using the formula:
Margin of error = Z * √ [(p * (1 - p)) / n]
Where:
Z is the z-value corresponding to the desired confidence level (98% confidence level corresponds to a z-value of 2.33, assuming a normal distribution).
p is the sample proportion.
n is the sample size.
Margin of error = 2.33 * √ [(0.249 * (1 - 0.249)) / 2404]
Margin of error ≈ 0.031
Now, we can construct the confidence interval by subtracting and adding the margin of error from the sample proportion:
Lower confidence limit = Sample proportion - Margin of error
Upper confidence limit = Sample proportion + Margin of error
Lower confidence limit = 0.249 - 0.031
Lower confidence limit ≈ 0.218
Upper confidence limit = 0.249 + 0.031
Upper confidence limit ≈ 0.280
Therefore, the 98% confidence interval for the population proportion of tenth graders reading at or below the eighth grade level is approximately 0.218 to 0.280.
To construct a 98% confidence interval for the population proportion of tenth graders reading at or below the eighth grade level, we can use the formula:
Confidence Interval = Sample Proportion ± Margin of Error
First, we need to calculate the sample proportion:
Sample Proportion = (Number of Students Reading at or Below 8th Grade Level) / (Total Sample Size)
Sample Proportion = 601 / 2404
Sample Proportion = 0.24937655 (rounded to 8 decimal places)
Next, we need to calculate the margin of error:
Margin of Error = Critical Value * Standard Error
Standard Error = sqrt((Sample Proportion * (1 - Sample Proportion)) / Sample Size)
First, find the critical value for a 98% confidence interval. We can use a table or a calculator to find this value. The critical value for a 98% confidence interval is approximately 2.33.
Next, substitute the values into the formula:
Standard Error = sqrt((0.24937655 * (1 - 0.24937655)) / 2404)
Standard Error ≈ 0.01047196 (rounded to 8 decimal places)
Margin of Error = 2.33 * 0.01047196
Margin of Error ≈ 0.024354 (rounded to 6 decimal places)
Finally, we can construct the confidence interval:
Confidence Interval = 0.24937655 ± 0.024354
Confidence Interval ≈ (0.224, 0.274)
Therefore, based on the given sample data, we can be 98% confident that the true proportion of tenth graders reading at or below the eighth grade level lies somewhere between 0.224 and 0.274.