Point and Interval Estimates of the Population Mean
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Point estimate of the population mean is a single value that is used to estimate the true population mean. It is calculated using sample statistics such as the sample mean or the sample median.
Interval estimate of the population mean is a range of values that is used to estimate the true population mean. It is calculated using sample statistics such as the sample mean, the sample standard deviation, and the sample size. The most commonly used interval estimate is the confidence interval, which provides a range of values that is likely to contain the true population mean with a certain level of confidence.
The formula for calculating the confidence interval is:
CI = X̄ ± Zα/2 * (σ/√n)
where CI is the confidence interval, X̄ is the sample mean, Zα/2 is the critical value of the standard normal distribution corresponding to the desired level of confidence, σ is the population standard deviation (or the sample standard deviation if the population standard deviation is unknown), and n is the sample size.
For example, if a sample of size 100 has a sample mean of 50 and a sample standard deviation of 10, and we want to construct a 95% confidence interval for the population mean, the critical value of Zα/2 is 1.96. Therefore, the confidence interval is:
CI = 50 ± 1.96 * (10/√100) = 50 ± 1.96
The confidence interval is 48.04 to 51.96, which means that we are 95% confident that the true population mean falls within this range.
Interval estimate of the population mean is a range of values that is used to estimate the true population mean. It is calculated using sample statistics such as the sample mean, the sample standard deviation, and the sample size. The most commonly used interval estimate is the confidence interval, which provides a range of values that is likely to contain the true population mean with a certain level of confidence.
The formula for calculating the confidence interval is:
CI = X̄ ± Zα/2 * (σ/√n)
where CI is the confidence interval, X̄ is the sample mean, Zα/2 is the critical value of the standard normal distribution corresponding to the desired level of confidence, σ is the population standard deviation (or the sample standard deviation if the population standard deviation is unknown), and n is the sample size.
For example, if a sample of size 100 has a sample mean of 50 and a sample standard deviation of 10, and we want to construct a 95% confidence interval for the population mean, the critical value of Zα/2 is 1.96. Therefore, the confidence interval is:
CI = 50 ± 1.96 * (10/√100) = 50 ± 1.96
The confidence interval is 48.04 to 51.96, which means that we are 95% confident that the true population mean falls within this range.
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