A new chemical has been found to be present in the human blood stream, and a medical group would like to study the presence of this chemical in some samples of patients. The presence of the chemical in a patient is measured by a score representing the \'parts per billion\' in which that chemical appears in the blood. It is known that, on this scale, men have an average score of 809.1 and a standard deviation of 54. It is also known that women have an average score of 831.42 and a standard deviation of 22. An assistant in the medical team has been handed a sample of 100 scores. The assistant knows that all of the scores are from one of the two genders, but the sample was not documented very well and so they do not which gender this is. Within the sample, the mean score is 821.52. Complete the following statements. Give your answers to 1 decimal place. If the sample came from a group of 100 men, then the sample mean is the standard deviations above the mean of the sampling distribution. In contrast, if the sample came from a group of 100 women, then the sample mean is standard deviations below the mean of the sampling distribution. Based on this, the assistant is more confident that the sample came from a group of 100
Note that the z score for sample means is
z = (x - u) * sqrt(n) / sigma
Thus, if it were from men,
z (men) = (821.52-809.1)*sqrt(100)/54 = 1.119230769 = 2.3
If it were from women,
z (women) = (821.52-831.42)*sqrt(100)/22 = -6.137931034 = -4.5
Thus,
a)
If the sample came from a group of 100 men, then the sample mean is 2.3 standard deviations ABOVE the mean of the sampling distribution. In contrast, if the sample came from a group of 100 women, then the sample mean is 4.5 standard deviations BELOW the mean of the sampling distribution. [answer]
b)
The less standard deviation from the mean, the more usual it is. Actually, 4.2 is a very unusual z score already.
Thus, Based on this, the assistant is more confident that the sample came from a group of 100 MEN. [ANSWER]
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