The heights of a particular breed of horse are normally distributed with a standard deviation of 6.5cm. The mean is unknown but is somewhere between 140cm and 150cm. Which one of the following statements cannot be true?Group of answer choicesThe median of the distribution is 148cmA large horse of this breed could plausibly reach a height of 163cmAbout 5% of horses of this breed have heights outside the range 130cm to 156cm.About 5% of horses of this breed have heights outside the range 142cm to 155cm.
Question
The heights of a particular breed of horse are normally distributed with a standard deviation of 6.5cm. The mean is unknown but is somewhere between 140cm and 150cm. Which one of the following statements cannot be true?Group of answer choicesThe median of the distribution is 148cmA large horse of this breed could plausibly reach a height of 163cmAbout 5% of horses of this breed have heights outside the range 130cm to 156cm.About 5% of horses of this breed have heights outside the range 142cm to 155cm.
Solution
In a normal distribution, the mean, median, and mode are all equal. Therefore, if the mean is somewhere between 140cm and 150cm, the median must also be within this range.
Statement 1: The median of the distribution is 148cm. This could be true, as 148cm is within the range of 140cm to 150cm.
Statement 2: A large horse of this breed could plausibly reach a height of 163cm. This could be true, as 163cm is within two standard deviations of the mean (since the standard deviation is 6.5cm, two standard deviations would be 13cm, and 150cm + 13cm = 163cm).
Statement 3: About 5% of horses of this breed have heights outside the range 130cm to 156cm. This could be true, as this range is within three standard deviations of the mean (since the standard deviation is 6.5cm, three standard deviations would be 19.5cm, and both 140cm - 19.5cm = 120.5cm and 150cm + 19.5cm = 169.5cm are outside the range 130cm to 156cm).
Statement 4: About 5% of horses of this breed have heights outside the range 142cm to 155cm. This statement cannot be true. In a normal distribution, about 95% of the data falls within two standard deviations of the mean. If the standard deviation is 6.5cm, then two standard deviations would be 13cm. Therefore, the range that includes about 95% of the data would be from the mean minus 13cm to the mean plus 13cm. Since the mean is between 140cm and 150cm, the range that includes about 95% of the data would be wider than 142cm to 155cm. Therefore, it cannot be true that about 5% of horses of this breed have heights outside the range 142cm to 155cm.
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