A random sample of 50 four-year-olds attending day care centers provided a yearly tuition average of 3987, with a population standard deviation of 630. Find the 90 percent confidence interval of the true mean. What is the upper bound of this interval? (Rounded to nearest whole number)
Z = (score-mean)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability = .05 to get the Z score.
Insert data into first equation to get the score.
To find the 90% confidence interval for the true mean, we need to use the formula:
Confidence Interval = X̄ ± Z * (σ / √n)
Where:
X̄ is the sample mean (3987 in this case)
Z is the Z-score for the desired confidence level (90% in this case)
σ is the population standard deviation (630 in this case)
n is the sample size (50 in this case)
Now, let's calculate the confidence interval step by step:
1. Find the Z-score for the 90% confidence level. The Z-score can be found using a Z-table or a calculator. For a 90% confidence level, the Z-score is approximately 1.645.
2. Substitute the values into the formula:
Confidence Interval = 3987 ± 1.645 * (630 / √50)
3. Calculate the standard error (σ / √n):
Standard error = 630 / √50
4. Substitute the values again:
Confidence Interval = 3987 ± 1.645 * (standard error)
5. Calculate the upper bound of the interval:
Upper bound = 3987 + 1.645 * (standard error)
Now, let's calculate the upper bound:
Standard error = 630 / √50 = 88.929
Upper bound = 3987 + 1.645 * 88.929 ≈ 3987 + 146.246 ≈ 4133
Therefore, the upper bound of the 90% confidence interval is approximately 4133 (rounded to the nearest whole number).
To find the confidence interval of the true mean, we can use the formula:
Confidence Interval = X̄ ± Z * (σ/√n)
where:
X̄ = sample mean
Z = Z-value (corresponding to the desired confidence level)
σ = population standard deviation
n = sample size
Given:
X̄ = 3987
σ = 630
n = 50
To find the Z-value corresponding to a 90% confidence level, we need to find the critical value from the Z-table or by using a statistical calculator. For a 90% confidence level, the Z-value is approximately 1.645.
Now, let's calculate the confidence interval:
Confidence Interval = 3987 ± 1.645 * (630/√50)
Confidence Interval = 3987 ± 1.645 * (630/7.071)
Confidence Interval = 3987 ± 1.645 * 89.002
Confidence Interval = 3987 ± 146.474
Confidence Interval ≈ (3839, 4135)
The confidence interval for the true mean tuition is approximately (3839, 4135).
To find the upper bound of this interval, we need to round up the upper limit:
Upper bound ≈ 4135 (rounded to the nearest whole number)
Therefore, the upper bound of the 90% confidence interval for the true mean is 4135.