Build a 95% confidence interval for the mean of the population from the given sample data. Assume that the population has a normal distribution.
16.4, 15.7, 16.2, 15.8 17.1
15.8, 15.9, 16.0, 16.4 15.0
*please help me solve this! I have no clue what I am supposed to do
Mean
Standard deviation
mean -+ sdv * z/a/2 / sqrt(n))
To build a 95% confidence interval for the mean of the population, you need to follow these steps:
Step 1: Calculate the sample mean.
Add up all the values in the sample data and divide the sum by the total number of observations. Given the sample data:
16.4, 15.7, 16.2, 15.8, 17.1, 15.8, 15.9, 16.0, 16.4, 15.0
The sum is 162.3, and there are 10 observations. Dividing the sum by 10 gives us a sample mean of 16.23.
Step 2: Calculate the sample standard deviation.
First, calculate the variance by finding the squared differences between each observation and the sample mean, summing them up, and dividing by the total number of observations minus 1 (n-1). Then, take the square root of the variance to find the sample standard deviation.
Using the same sample data, the variance is 0.14, and the sample standard deviation is the square root of the variance, which is approximately 0.37.
Step 3: Determine the critical value.
The critical value corresponds to the desired confidence level and the size of your sample. In this case, we are looking for a 95% confidence interval, which means we want to find the critical value that leaves 2.5% of the probability in each tail of the distribution. Since we have a small sample size (<30), we need to use a t-distribution instead of a standard normal distribution.
With a sample size of 10, the degrees of freedom (df) are 10-1 = 9. The critical value can be found using a t-table or a statistical software. For a 95% confidence level and 9 degrees of freedom, the critical value is approximately 2.262.
Step 4: Calculate the margin of error.
The margin of error represents the range around the sample mean within which the population mean is likely to fall, given the desired confidence level. It is calculated by multiplying the critical value from step 3 by the sample standard deviation divided by the square root of the sample size.
With the critical value of 2.262 and a sample standard deviation of 0.37, and a sample size of 10, the margin of error is approximately 0.263.
Step 5: Calculate the confidence interval.
The confidence interval is constructed by subtracting the margin of error from the sample mean to get the lower bound, and adding the margin of error to the sample mean to get the upper bound.
Using the sample mean of 16.23 and the margin of error of 0.263, the confidence interval is (15.967, 16.493).
Therefore, the 95% confidence interval for the mean of the population is (15.967, 16.493).