A random sample of size 15 taken from normally distributed population revealed a sample mean of 75 and a sampl variance of 25. The lower limit of 95% confidence interval for the population mean would equal
75%
To calculate the lower limit of a 95% confidence interval for the population mean, you can use the formula:
Lower Limit = Sample Mean - (Critical Value * Standard Error)
1. Find the critical value corresponding to a 95% confidence level. For a sample size of 15, the critical value can be found using a t-distribution table or a statistical calculator. Since the sample size is small (n < 30), the t-distribution should be used. For a 95% confidence level and degrees of freedom (df) = n-1 = 14, the critical value is approximately 2.145.
2. Calculate the standard error (SE), which is the standard deviation of the sample divided by the square root of the sample size. Since the sample variance is given as 25, the sample standard deviation (s) is the square root of 25, which is 5. Therefore, the standard error (SE) can be calculated as follows:
SE = s / √n
SE = 5 / √15
3. Calculate the lower limit for the confidence interval using the formula:
Lower Limit = Sample Mean - (Critical Value * Standard Error)
Lower Limit = 75 - (2.145 * (5 / √15))
After plugging in the values, calculate the lower limit to find the answer.
To find the lower limit of the 95% confidence interval for the population mean, we can use the formula:
Lower Limit = Sample Mean - Margin of Error
First, we need to calculate the margin of error, which depends on the sample size, sample standard deviation, and the confidence level. Since we have the sample variance but not the sample standard deviation, we can calculate it using the formula:
Sample Standard Deviation = square root of Sample Variance
In this case, the sample variance is 25, so the sample standard deviation is the square root of 25, which is 5.
Next, we calculate the margin of error using the formula:
Margin of Error = (Critical Value) * (Standard Deviation / square root of Sample Size)
The critical value represents the number of standard deviations we need to go from the mean to capture a certain percentage of the population. For a 95% confidence level, we use a critical value corresponding to the z-score of 1.96.
Finally, we can substitute the values into the formula:
Margin of Error = 1.96 * (5 / square root of 15)
Calculating this expression, we get the margin of error.
Now, we can find the lower limit of the confidence interval by subtracting the margin of error from the sample mean:
Lower Limit = Sample Mean - Margin of Error
Substituting the values, we get the lower limit of the 95% confidence interval for the population mean.
Recall that 95% interval is µ±1.96σ/√n
so plug in your numbers.