Question

suppose the population of iq scores in the town or city where you live is bell - shaped, with a mean of 102 and a standard deviation of 15. describe the frequency curve for possible sample means that would result from random samples of 100 iq scores. using the central limit theorem, we would expect the possible sample means to be bell - shaped, with a mean of and a standard deviation of. therefore, we would find them: between and about 68% of the time, between and about 95% of the time, between and about 99.7% of the time.

Explanation:

Step1: Mean of sample means

The mean of the sampling - distribution of the sample mean ($\mu_{\bar{x}}$) is equal to the population mean ($\mu$). Given $\mu = 102$, so $\mu_{\bar{x}}=102$.

Step2: Standard deviation of sample means

The standard deviation of the sampling - distribution of the sample mean (also known as the standard error, $\sigma_{\bar{x}}$) is calculated by the formula $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. Given $\sigma = 15$ and $n = 100$, then $\sigma_{\bar{x}}=\frac{15}{\sqrt{100}}=\frac{15}{10}=1.5$.

Step3: 68% confidence interval

For a normal distribution, about 68% of the data lies within 1 standard deviation of the mean. So the interval is $\mu_{\bar{x}}\pm\sigma_{\bar{x}}$, which is $102 - 1.5=100.5$ and $102 + 1.5 = 103.5$.

Step4: 95% confidence interval

For a normal distribution, about 95% of the data lies within 2 standard deviations of the mean. So the interval is $\mu_{\bar{x}}\pm2\sigma_{\bar{x}}$, which is $102-2\times1.5 = 99$ and $102 + 2\times1.5=105$.

Step5: 99.7% confidence interval

For a normal distribution, about 99.7% of the data lies within 3 standard deviations of the mean. So the interval is $\mu_{\bar{x}}\pm3\sigma_{\bar{x}}$, which is $102-3\times1.5 = 97.5$ and $102+3\times1.5 = 106.5$.

Answer:

The mean of the possible sample means is 102, the standard deviation is 1.5. We would find them: between 100.5 and 103.5 about 68% of the time, between 99 and 105 about 95% of the time, between 97.5 and 106.5 about 99.7% of the time.