The IQ test scores of 74 seventh-grade girls in a midwestern school district are given. 97 115 111 112 118 113 124 127 119 97 102 110 110 107 105 105 89 102 91 114 132 111 128 118 112 107 103 Test Scores 104 106 113 136 106 123 120 103 115 110 90 114 114 103 105 119 86 107 96 112 112 109 113 124 126 93 123 106 114 111 108 100 111 93 128 128 116 127 119 110 100 104 130 120 103 107 98 Click to download the data in your preferred format. Estimate the mean IQ score for all seventh-grade girls in the school district, using a 98% confidence
The Correct Answer and Explanation is:
Correct Answer:
The estimated mean IQ score is 110.73, with a 98% confidence interval of (107.73, 113.73).
Explanation:
To estimate the average IQ score of all seventh-grade girls in a school district, we use a confidence interval based on a random sample. In this case, the sample includes 74 girls, and their IQ scores are provided.
A confidence interval gives a range of values within which we expect the true population mean to lie. Because we do not know the actual IQ average of every seventh-grade girl in the district, we use the data from the sample to make an informed estimate.
We begin by calculating the sample mean, which is the average of all IQ scores in the sample. This value is 110.73. Next, we calculate the sample standard deviation, which measures how much individual IQ scores differ from the average.
Since the population standard deviation is unknown, we use the t-distribution instead of the normal (z) distribution. The t-distribution is more appropriate for smaller sample sizes and provides more accurate interval estimates.
Using a 98% confidence level, we find the corresponding t-value from the t-distribution table for 73 degrees of freedom (n – 1). This t-value helps us determine the margin of error, which is added to and subtracted from the sample mean to get the confidence interval.
The result is a confidence interval from 107.73 to 113.73. This means we are 98% confident that the true average IQ score for all seventh-grade girls in the district falls within this range.
This method is widely used in statistics to make population-level estimates from sample data. It balances the need for accuracy with the reality that testing an entire population is rarely possible.
