A psychologist is interested in the mean IQ score of a given group of children. It is known that the IQ scores of the group have a standard deviation of 16. The psychologist randomly selects 80 children from this group and finds that their mean IQ score is 110. Based on this sample, find a 90% confidence interval for the true mean IQ score for all children of this group. Then give its lower limit and upper limit.Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place. (If necessary, consult a list of formulas.)Lower limit: Upper limit:
Question
A psychologist is interested in the mean IQ score of a given group of children. It is known that the IQ scores of the group have a standard deviation of 16. The psychologist randomly selects 80 children from this group and finds that their mean IQ score is 110. Based on this sample, find a 90% confidence interval for the true mean IQ score for all children of this group. Then give its lower limit and upper limit.Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place. (If necessary, consult a list of formulas.)Lower limit: Upper limit:
Solution
To calculate the confidence interval, we first need to find the standard error. The standard error (SE) is calculated by dividing the standard deviation (SD) by the square root of the sample size (n).
In this case, the standard deviation is 16 and the sample size is 80.
SE = SD / sqrt(n) SE = 16 / sqrt(80) SE = 16 / 8.944 SE = 1.789
Next, we need to find the z-score for a 90% confidence interval. The z-score for a 90% confidence interval is 1.645 (you can find this in a standard z-table or use a calculator).
Now we can calculate the margin of error (ME). The margin of error is calculated by multiplying the standard error by the z-score.
ME = SE * z ME = 1.789 * 1.645 ME = 2.942
Finally, we can calculate the confidence interval. The confidence interval is calculated by subtracting the margin of error from the sample mean for the lower limit and adding the margin of error to the sample mean for the upper limit.
Lower limit = sample mean - ME Lower limit = 110 - 2.942 Lower limit = 107.058
Upper limit = sample mean + ME Upper limit = 110 + 2.942 Upper limit = 112.942
So, the 90% confidence interval for the true mean IQ score for all children of this group is between 107.1 and 112.9 (rounded to one decimal place).
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