Mean IQ of Data Scientists See the preceding exercise, in whic...

Question

Mean IQ of Data Scientists See the preceding exercise, in which we can assume that sigma=15 for the IQ scores. Data scientists are a group with IQ scores that vary less than the IQ scores of the general population. Find the sample size needed to estimate the mean IQ of data scientists, given that we want 98% confidence that the sample mean is within 2 IQ points of the population mean. Does the sample size appear to be practical?

Accepted Answer

Below there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A psychologist wants to estimate the mean reaction time in milliseconds for a cognitive task. The population standard deviation is known to be 8 milliseconds. What sample size is needed to ensure that the sample mean is within 1.5 milliseconds of the population mean with 95% confidence? Awesome. So it appears for this particular prong we're ultimately trying to figure out what sample size is required or needed in order to ensure that the sample mean is within 1.5 milliseconds of the population mean with a 95% confidence. So that's our final answer we're ultimately trying to solve for. So with that said, let's read off our multiple choice answers to see what our final answer might be. A is 105, B is 110, C is 115, and D is 120. Awesome. So first off, in order to determine the required sample size, we must first recall and use the formula for sample size when estimating a population mean with a known population standard deviation, and this equation states that N is equal to parentheses. Capital Z multiplied by sigma, divided by E. All to the power of 2. Where n to be exact, is the required sample size, Z is equal to our Z value, which corresponds to the desired confidence level, which in this case is 95% confidence, which should be, let's make a note of this off to the side, so that will mean that Z is equal to 1.96. Fantastic for this particular prompt. And Sigma is our population standard deviation, which is 8 milliseconds for this particular prom, and E is the margin of error, and the margin of error for this particular prom is 1.5 milliseconds. So with that said, let us now plug in all of our known variables and solve for N. So N is going to be equal to parentheses 1.96 multiplied by 8 divided by 1.5, all to the power 2, so we plug that into our calculator. We will find, or rather, let's make this simple. So when we calculate the numerator first, we will find that the numerator is 1.96 multiplied by 8, which is going to be equal to 15.68. Fantastic. And then if we divide by the margin of error, that would mean that we will take 15.68 divided by 1.5, which will give us an answer of 10.45. And if we square the results, that will mean that 10.45 to the power of 2 will give us an answer of 109.2, which is approximately equal to 110 when we round to the nearest whole number, and that is our final answer. Hooray, we did it. So therefore, The required sample size is approximately 110 because we need to round up to the next whole number. So therefore, without a doubt, the this particular psychologist would need a sample size of 110. To ensure that the sample mean will be within 1.5 milliseconds of the population mean with a 95% confidence. So that means without a doubt, our final answer has to be multiple choice answer letter B, which once again is 110. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Confidence Interval

Sample Size Calculation

Standard Deviation

Watch next