A sample of n = 9 has a mean of M = 72 with SS = 162. Use the sample to make a point estimate of ì and to construct the 90% confidence interval for U. (Round all answers to two decimal places.)
U = ? ± ? ( ?)
The 90% confidence interval for U:
( ? , ? )
SS/n = variance = SD^2
90% = mean ± 1.645 SD
To make a point estimate of µ (population mean) based on the sample, we can use the sample mean. In this case, the sample mean (M) is given as 72. So, our point estimate for µ is 72.
To construct the 90% confidence interval for µ, we need to calculate the margin of error and then determine the lower and upper bounds.
1. Calculate the margin of error (ME):
The margin of error depends on the sample size (n) and the standard deviation (SD) of the population. However, in this case, we are not given the population standard deviation. Instead, we are provided with the sum of squares (SS) which can be used to calculate the sample variance (s^2).
2. Calculate the sample variance (s^2):
The sample variance is calculated as the sum of squares divided by the degrees of freedom (n-1). In this case, the SS is given as 162 and the sample size is 9, so we have:
s^2 = SS / (n - 1) = 162 / (9 - 1) = 20.25
3. Calculate the standard error (SE):
The standard error is the square root of the sample variance divided by the square root of the sample size. In this case:
SE = sqrt(s^2 / n) = sqrt(20.25 / 9) = sqrt(2.25) = 1.5
4. Determine the critical value (z):
For a 90% confidence interval, we need to find the z-value that corresponds to a central area of 0.90. This can be done using a standard normal distribution table or a calculator. The critical value for a 90% confidence interval is approximately 1.645.
5. Calculate the margin of error (ME):
The margin of error is determined by multiplying the critical value (z) by the standard error (SE). In this case:
ME = z * SE = 1.645 * 1.5 = 2.47
6. Calculate the lower and upper bounds:
The lower bound is calculated by subtracting the margin of error from the point estimate (sample mean), and the upper bound is calculated by adding the margin of error to the point estimate.
Lower bound = 72 - 2.47 = 69.53
Upper bound = 72 + 2.47 = 74.47
Therefore, the 90% confidence interval for µ is (69.53, 74.47).
To summarize:
- Point estimate of µ: 72
- 90% confidence interval for µ: (69.53, 74.47)