A simple random sample of 25 has been collected from a normally distributed population for which it is know that o=17.0. The sample mean has been calculated as 342.0, and the sample standard deviation is s=14.9. Construct and interpret the 95% and 99% confidence intervals for the population mean.
To construct confidence intervals for the population mean, we can use the formula:
Confidence Interval = sample mean ± (Critical value * (sample standard deviation / square root of sample size))
First, let's calculate the critical values for the 95% confidence level and the 99% confidence level. The critical values represent the number of standard errors away from the mean that captures a certain percentage of the population.
For a 95% confidence level, the critical value can be found by looking up the z-value in the standard normal distribution table. We divide the significance level (5%) by 2 to account for both tails of the distribution. The resulting z-value is approximately 1.96.
For a 99% confidence level, the critical value can be found in a similar manner. Dividing the significance level (1%) by 2 gives a z-value of approximately 2.58.
Now we can substitute the given values into the formula:
For the 95% confidence interval:
Lower bound = 342.0 - (1.96 * (14.9 / sqrt(25)))
Upper bound = 342.0 + (1.96 * (14.9 / sqrt(25)))
For the 99% confidence interval:
Lower bound = 342.0 - (2.58 * (14.9 / sqrt(25)))
Upper bound = 342.0 + (2.58 * (14.9 / sqrt(25)))
Let's calculate these confidence intervals:
For the 95% confidence interval:
Lower bound = 342.0 - (1.96 * (14.9 / sqrt(25))) ≈ 332.92
Upper bound = 342.0 + (1.96 * (14.9 / sqrt(25))) ≈ 351.08
For the 99% confidence interval:
Lower bound = 342.0 - (2.58 * (14.9 / sqrt(25))) ≈ 330.75
Upper bound = 342.0 + (2.58 * (14.9 / sqrt(25))) ≈ 353.25
Interpretation:
For the 95% confidence interval, we can say that we are 95% confident that the population mean lies between approximately 332.92 and 351.08.
For the 99% confidence interval, we can say that we are 99% confident that the population mean lies between approximately 330.75 and 353.25.
These intervals represent the range within which we estimate the true population mean lies, with a certain level of confidence.