The mean of a population being sampled is 64, and the standard deviation is 6. If the sample size is 50, what is the standard error of the mean? (Round off your answer to the nearest hundredth.)
Answer (2)
Calculate the standard error of the mean using the formula: SEM = n σ .
Substitute the given values: σ = 6 and n = 50 , so SEM = 50 6 .
Calculate the square root and divide: SEM ≈ 7.071 6 ≈ 0.8485 .
Round the result to the nearest hundredth: 0.85 .
Explanation
Understand the problem and provided data We are given a population with a mean of 64 and a standard deviation of 6. We are taking a sample of size 50 from this population. Our goal is to find the standard error of the mean.
Recall the formula for standard error of the mean The standard error of the mean (SEM) is a measure of the statistical accuracy of an estimate of a population mean. It is calculated by dividing the population standard deviation by the square root of the sample size. The formula is: SEM = n σ where σ is the population standard deviation and n is the sample size.
Plug in the values In this problem, the population standard deviation σ = 6 and the sample size n = 50 . Plugging these values into the formula, we get: SEM = 50 6 Now, we calculate the square root of 50: 50 ≈ 7.071
Calculate the standard error Now, we divide the population standard deviation by the square root of the sample size: SEM = 7.071 6 ≈ 0.8485 We are asked to round the answer to the nearest hundredth.
Round to the nearest hundredth Rounding 0.8485 to the nearest hundredth gives us 0.85. Therefore, the standard error of the mean is approximately 0.85.
Examples
Understanding the standard error of the mean is crucial in various real-world scenarios. For instance, in clinical trials, researchers often take samples of patients to test the effectiveness of a new drug. The standard error helps them determine how accurately the sample mean represents the entire patient population. A smaller standard error indicates a more precise estimate of the population mean, allowing for more confident conclusions about the drug's efficacy. Similarly, in market research, the standard error helps assess the reliability of survey results when estimating the average customer satisfaction or product preference.
The standard error of the mean (SEM) is calculated using the formula SEM = n σ . With a population standard deviation of 6 and a sample size of 50, the SEM is approximately 0.85 after rounding to the nearest hundredth. The SEM provides insight into how close our sample mean is likely to be to the true population mean.
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