Find The Mean And Standard Error Of The Sample Means

Find The Mean And Standard Error Of The Sample Means That Is Normal

Find the mean and standard error of the sample means that is normally distributed with a mean of 10 and a standard deviation of 2, given a sample size of 25. Using this distribution, find the probability of a score less than 11.

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Understanding the characteristics of sample distributions is fundamental in inferential statistics. When dealing with a normally distributed population, the sample mean itself follows a normal distribution, known as the sampling distribution of the mean. Calculating the mean and standard error of this sampling distribution enables researchers to make probabilistic statements about the data, including estimating the likelihood of observing specific sample means.

Given the population mean (μ) of 10 and a population standard deviation (σ) of 2, with a sample size (n) of 25, the sampling distribution of the mean will also be normally distributed. The mean of this sampling distribution remains the same as the population mean, which is μ = 10. Computing the standard error (SE) involves dividing the population standard deviation by the square root of the sample size:

SE = σ / √n = 2 / √25 = 2 / 5 = 0.4.

This standard error signifies the standard deviation of the sampling distribution of the mean, reflecting how much the sample means are expected to vary from the population mean due to sampling variability.

Next, the problem asks to find the probability of obtaining a sample mean less than 11. To approach this, we convert the sample mean to a z-score, which standardizes the value relative to the sampling distribution:

z = (X̄ - μ) / SE = (11 - 10) / 0.4 = 1 / 0.4 = 2.5.

Using the standard normal distribution table or calculator, the probability corresponding to a z-score of 2.5 is approximately 0.9938. This indicates that there is a 99.38% probability that the sample mean will be less than 11, assuming the population parameters and sample size remain constant.

Understanding these calculations underscores the importance of standard error in inferential statistics, as it allows researchers to determine the likelihood of observed sample statistics under the assumption of known population parameters. Such insights are critical for hypothesis testing, confidence interval construction, and decision-making in various fields, including psychology, education, and social sciences.

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