Past experience has shown that scores on a particular skills-based test (A) are distributed Normally (mean=25, std=8) and scores on an alternative skills-based test (B) are distributed Normally (mean=30, std=6).The correlation (r) between the two test scores is zero. Which of the follwing statements is FALSE?Group of answer choicesThe two tests must be testing unrelated skills.Let T= total score for a randomly selected individual: T = A + B. It is expected that approximately 95% of total scores (T) will be between 35 and 75.There is insufficient information available to determine if A and B are independent (i.e. insufficient information to determine if scores on one test affect an individuals score on the other test.)Total score (T=A + B) can reasonably be modelled using a normal distribution.Approximately 90% of the total scores (T) are below 67.8.
Question
Past experience has shown that scores on a particular skills-based test (A) are distributed Normally (mean=25, std=8) and scores on an alternative skills-based test (B) are distributed Normally (mean=30, std=6).The correlation (r) between the two test scores is zero. Which of the follwing statements is FALSE?Group of answer choicesThe two tests must be testing unrelated skills.Let T= total score for a randomly selected individual: T = A + B. It is expected that approximately 95% of total scores (T) will be between 35 and 75.There is insufficient information available to determine if A and B are independent (i.e. insufficient information to determine if scores on one test affect an individuals score on the other test.)Total score (T=A + B) can reasonably be modelled using a normal distribution.Approximately 90% of the total scores (T) are below 67.8.
Solution
Given that the scores on tests A and B are normally distributed with means of 25 and 30, and standard deviations of 8 and 6, respectively, and that the correlation between the two test scores is zero, let's evaluate each of the options:
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The two tests must be testing unrelated skills. - FALSE. A zero correlation simply means that there is no linear relationship between the scores on the two tests. It does not necessarily mean that the skills tested are unrelated.
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Let T= total score for a randomly selected individual: T = A + B. It is expected that approximately 95% of total scores (T) will be between 35 and 75. - TRUE. The sum of two normally distributed random variables is also normally distributed, with mean equal to the sum of the means and variance equal to the sum of the variances. Therefore, the mean of T is 25 + 30 = 55, and the standard deviation of T is sqrt(8^2 + 6^2) = 10. Approximately 95% of observations from a normal distribution fall within two standard deviations of the mean, so approximately 95% of total scores will be between 55 - 210 = 35 and 55 + 210 = 75.
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There is insufficient information available to determine if A and B are independent (i.e. insufficient information to determine if scores on one test affect an individuals score on the other test.) - TRUE. Correlation and independence are not the same thing. A zero correlation does not imply independence.
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Total score (T=A + B) can reasonably be modelled using a normal distribution. - TRUE. The sum of two normally distributed random variables is also normally distributed.
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Approximately 90% of the total scores (T) are below 67.8. - TRUE. The 90th percentile of a normal distribution is approximately 1.28 standard deviations above the mean, so the 90th percentile of T is 55 + 1.28*10 = 67.8.
Therefore, the statement "The two tests must be testing unrelated skills." is FALSE.
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