{"id":509,"date":"2022-07-11T19:49:27","date_gmt":"2022-07-11T19:49:27","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/alphamodule\/?post_type=chapter&p=509"},"modified":"2022-07-11T19:49:27","modified_gmt":"2022-07-11T19:49:27","slug":"z-score-and-the-empirical-rule-apply-it-3","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/alphamodule\/chapter\/z-score-and-the-empirical-rule-apply-it-3\/","title":{"raw":"Z-Score and the Empirical Rule: Apply It 3","rendered":"Z-Score and the Empirical Rule: Apply It 3"},"content":{"raw":"

Z-Scores<\/h3>\r\nA higher organ weight is an indicator of higher toxicity. Suppose a mouse has a liver weight of\u00a0[latex]1.07[\/latex] g and a spleen weight of\u00a0[latex]0.104[\/latex] g. Is either of these values extreme?\u00a0 How many standard deviations from the mean do these values lie, and in what direction? We can use the z-score for each of these values to help us answer these questions. In the following questions, calculate the z-score for these weights then interpret that score. Remember, the z-score is a\u00a0number of standard deviations<\/em>, and has no units associated with it. It only gives relative proximity (distance and direction) from the mean of a quantitative variable.\r\n
\r\n

recall<\/h3>\r\nIn the following questions, you'll need to calculate and interpret z-scores. Take a moment to refresh the formula if needed.\r\n\r\nCore skill: [reveal-answer q=\"785224\"]Express the formula obtaining a z-score[\/reveal-answer]\r\n[hidden-answer a=\"785224\"][latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x[\/latex] is an observation value, [latex]\\mu[\/latex] is the population mean, and [latex]\\sigma[\/latex] is the standard deviation[\/hidden-answer]\r\n\r\n<\/div>\r\n
\r\n

question 5<\/h3>\r\n[ohm_question hide_question_numbers=1]241235[\/ohm_question]\r\n\r\n[reveal-answer q=\"360167\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"360167\"]Use the formula for calculating z-scores and round your answer to the nearest thousandth. How many standard deviations from the mean is this weight?[\/hidden-answer]\r\n\r\n<\/div>\r\n
\r\n

question 6<\/h3>\r\n[ohm_question hide_question_numbers=1]241238[\/ohm_question]\r\n\r\n[reveal-answer q=\"674426\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"674426\"]Use the formula for calculating z-scores and round your answer to the nearest thousandth. How many standard deviations from the mean is this weight?[\/hidden-answer]\r\n\r\n<\/div>\r\n
\r\n

question 7<\/h3>\r\n[ohm_question hide_question_numbers=1]241241[\/ohm_question]\r\n\r\n[reveal-answer q=\"391619\"]Hint[\/reveal-answer]\r\n[hidden-answer a=\"391619\"]Which organ is relatively heavier than a typical weight? Put this in the context of the Empirical Rule.[\/hidden-answer]\r\n\r\n<\/div>\r\n
\r\n

video placement<\/h3>\r\n[wrap-up video: In the final question of this activity, you compared two organ weights, one liver and one spleen, to determine which had a higher level of toxicity. But the distribution for liver and spleen weights didn't have the same mean, so simply comparing one weight to the other wouldn't help. Mouse spleens are naturally much lighter than mouse livers. You needed to compare their \"unusualness\" instead. To do so, you calculated z-scores for each weight. This let you determine which of the two was further from the mean weight for all such mouse organs [voice over the Empirical graph again here], which let you know which of the two was relatively heavier for it's type. Remember that by calculating the z-score, you are calculating a distance in the distribution, not a weight in grams. Z-scores have no units associated with them. You found that the spleen showed a higher level of toxicity because the weight of the spleen was unusual, at 2.571 standard deviations above the mean. The weight of the liver, by contrast, was only 0.816 standard deviations away, within the middle 68% of all mouse liver weights.\"]<\/span>\r\n\r\n<\/div>","rendered":"

Z-Scores<\/h3>\n

A higher organ weight is an indicator of higher toxicity. Suppose a mouse has a liver weight of\u00a0[latex]1.07[\/latex] g and a spleen weight of\u00a0[latex]0.104[\/latex] g. Is either of these values extreme?\u00a0 How many standard deviations from the mean do these values lie, and in what direction? We can use the z-score for each of these values to help us answer these questions. In the following questions, calculate the z-score for these weights then interpret that score. Remember, the z-score is a\u00a0number of standard deviations<\/em>, and has no units associated with it. It only gives relative proximity (distance and direction) from the mean of a quantitative variable.<\/p>\n

\n

recall<\/h3>\n

In the following questions, you’ll need to calculate and interpret z-scores. Take a moment to refresh the formula if needed.<\/p>\n

Core skill: <\/p>\n

Express the formula obtaining a z-score<\/span><\/p>\n
[latex]z=\\dfrac{x-\\mu}{\\sigma}[\/latex], where [latex]x[\/latex] is an observation value, [latex]\\mu[\/latex] is the population mean, and [latex]\\sigma[\/latex] is the standard deviation<\/div>\n<\/div>\n<\/div>\n
\n

question 5<\/h3>\n