In Exercises 3–6, construct the indicated confidence interval ...

Question

In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?

c=0.95, x̅=3.46, s=1.63, n=16

Accepted Answer

Hello 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 researcher measures the reaction time in units of seconds of 10 participants in a cognitive test. The sample mean reaction time is 0.82 seconds with a sample standard deviation of 0.13 seconds. Assuming that the reaction times are normally distributed, construct a 99% confidence interval for the population mean reaction time. Awesome. So it appears for this particular problem we're asked to assume that the reaction times are normally distributed and we're ultimately trying to construct a 99% confidence interval for the population mean reaction time. So ultimately our final answer we're trying to solve for is we're trying to create this confidence interval. Or rather determine this confidence interval. So with that in mind, our first step that we need to take in order to solve this particular problem is we need to identify the sample statistics and the confidence level. So we're told that the sample mean, which will denote as as X bar, and once again our sample mean X bar for this particular problem is equal to 0.82 seconds. And the sample standard deviation, which we'll call S in this case, is equal to 0.13 seconds, and the sample size we're told is 10 participants, so that will mean that N is equal to 10. And we're also told that the confidence interval, which we'll call CI, is equal to 99%. Fantastic. So now we need to note that since the sample size is small because N is less than 30 and the population standard deviation is unknown, we need to recall and use the T distribution. So in order to find the appropriate T value for a 99% confidence interval with degrees of freedom, DF is equal to N minus 1, which is equal to 10 minus 1, which is equal to 9. That will mean for a 99% confidence and a DF is equal to 9, the critical T value will be approximately T power of star is equal to 3.250. So with that in mind, we should recall it and use the equation to help us calculate the standard error of the mean, which will denote as SEM for short. So SEM, which once again is the standard error of the mean. is equal to S divided by the square root of N, which is equal to any plug in or known variables. 0.13 divided by the square root of 10. Fantastic. And when we move right along here, we now need to calculate the margin of error E. and as we should recall, the equation to determine the margin of error is E is equal to T to the power of star multiplied by SEM. Which is equal to when you plug in known variables 3.250 multiplied by 0.13 divided by the square root of 10. So when we plug in that expression into our calculator and we round the two decial places, so rounding the two decial places, we will get 0.13. So once again, E is approximately equal to 0.13. Fantastic. So now our next step is we need to construct the confidence interval, and the equation for the confidence interval, as we should recall, states that X bar plus or minus E will be equal to 0.82 plus or minus 0.13. Which will mean that we will get an answer of parentheses 0.82 minus 0.13. 0.82 plus 0.13, which will give us an answer of parentheses 0.69 0.95, and that is our final answer for the confidence interval. Hooray, we did it. So therefore, going back to look at the problem itself and to recap what we've learned, the 99% confidence interval for the population mean reaction time is, once again, 0.69 seconds because we do not want to forget our units. 0.95 seconds. So once again to make sure we know our final answer for the confidence interval is parentheses 0.69 seconds, 0.95 seconds. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?

c=0.95, x̅=3.46, s=1.63, n=16

Key Concepts

Confidence Interval

Normal Distribution

t-Distribution

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