a hair color manufacturer performed a survey which was normally distributed. they found that the average age…

a hair color manufacturer performed a survey which was normally distributed. they found that the average age at which a persons hair starts turning gray is 32 years, with a standard deviation of 4 years. which of the following graphs displays the normal distribution of the average age at which a persons hair starts turning gray?

Answer

Explanation:

Step1: Recall normal - distribution properties

In a normal distribution, the mean is at the center of the bell - shaped curve. Here, the mean age $\mu = 32$ years and the standard deviation $\sigma=4$ years. The values on the x - axis should be symmetrically spaced around the mean with intervals of the standard deviation.

Step2: Analyze option W

The values on the x - axis of option W are $24,28,32,36,40,44,48$. The mean is 32 (in the center), and the values are spaced at intervals of 4 (the standard deviation) around the mean. For example, $32 - 4=28$, $32+4 = 36$, $32 - 2\times4=24$, $32 + 2\times4=40$, etc.

Step3: Analyze option X

The values on the x - axis of option X are $20,24,28,32,36,40,44$. The mean is 32, but the left - most value 20 is 3 standard deviations ($3\times4 = 12$) away from the mean while the right - most value 44 is 3 standard deviations away. The spacing is not consistent with the standard deviation for all intervals around the mean as compared to the normal distribution properties.

Step4: Analyze option Y

The values on the x - axis of option Y are $23,26,29,32,35,38,41$. The mean is 32, but the values are spaced at intervals of 3, not 4 (the standard deviation).

Step5: Analyze option Z

The distribution in option Z is not symmetric around the mean. A normal distribution is symmetric about the mean, and this option does not follow that property.