random sample of 50 four year olds attending daycare centers provided a yearly tuition average of 3987$ and the population standard deviation of $630. Find the 90% confidence interval of the true mean
To find the 90% confidence interval of the true mean tuition for four-year-olds attending daycare centers, we can use the formula:
Confidence Interval = sample mean ± (Z * (sample standard deviation / √n))
where:
- sample mean is the average tuition of the random sample
- Z is the Z-score corresponding to the desired confidence level (in this case, 90% confidence level)
- sample standard deviation is the standard deviation of the random sample
- n is the sample size
Let's do the calculations step by step:
Step 1: Calculate the Z-score
For a 90% confidence level, the Z-score is 1.645. You can look up this value in a Z-table.
Step 2: Calculate the margin of error
The margin of error is calculated by multiplying the Z-score with the standard deviation divided by the square root of the sample size:
Margin of Error = 1.645 * (630 / √50)
Step 3: Calculate the confidence interval
The confidence interval is calculated by adding and subtracting the margin of error from the sample mean:
Confidence Interval = sample mean ± margin of error
Now we can calculate the confidence interval:
Step 4: Calculate the sample mean
Given that the sample mean is $3987, we can use this value in the formula.
Step 5: Calculate the confidence interval
Confidence Interval = $3987 ± (1.645 * (630 / √50))
Calculate the square root:
Confidence Interval = $3987 ± (1.645 * (630 / 7.071))
Simplify the expression:
Confidence Interval = $3987 ± (1.645 * 89.007)
Calculate the multiplication:
Confidence Interval = $3987 ± 146.342
Final result:
Confidence Interval ≈ $3839 to $4135
Therefore, we can say with 90% confidence that the true mean tuition for four-year-olds attending daycare centers falls within the range of approximately $3839 to $4135.
To find the 90% confidence interval of the true mean, you can use the formula:
Confidence Interval = Sample Mean +/- (Z * (Population Standard Deviation / sqrt(Sample Size)))
Here, the sample mean is $3987, the population standard deviation is $630, and the sample size is 50.
First, we need to determine the value of Z for a 90% confidence level. The Z-score corresponds to the desired confidence level and can be found using a Z-table or a statistical calculator. For a 90% confidence level, the corresponding Z-value is approximately 1.645.
Now we can calculate the confidence interval:
Confidence Interval = $3987 +/- (1.645 * ($630 / sqrt(50)))
Calculating the square root of 50, we get:
Confidence Interval = $3987 +/- (1.645 * ($630 / 7.071))
Simplifying further, we have:
Confidence Interval = $3987 +/- (1.645 * 89)
Confidence Interval = $3987 +/- 146.505
Therefore, the 90% confidence interval for the true mean lies between $3830.495 and $4143.505.