Question

which of the following statements are true: (a) if the standard deviation of a collection of data is zero then all the data in this collection are equal to each other (b) if you work for quality control in a company then your preference is that the measurements of your companys product have a large standard deviation (c) if a value is significantly low then the z - score of this value is significantly high (d) the mean of all house prices in lake worth is the best representative of typical house prices in lake worth (e) the 5 - number summary of a collection of data are the minimum, the mean, the standard deviation, the median, and the maximum of this collected data (f) the mean is not a resistant parameter (g) if population1 and population2 have the same standard deviation then population1 and population2 have the same variation amongst their data. a. (a), (g) and (b) b. (g), (e), and (f) c. (a) and (f) d. (a), (b), (g), and (d)

Explanation:

Step1: Analyze statement (a)

The formula for standard - deviation $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}}$. If $\sigma = 0$, then $\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=0$. This implies $x_{i}=\bar{x}$ for all $i$, so all data values are equal.

Step2: Analyze statement (b)

In quality control, a small standard deviation is preferred as it indicates consistency in product measurements. So, a large standard deviation is not a preference.

Step3: Analyze statement (c)

A significantly low value has a significantly low (negative) z - score, not a significantly high z - score. The z - score is calculated as $z=\frac{x-\mu}{\sigma}$.

Step4: Analyze statement (d)

The mean of house prices can be affected by outliers (very high or very low priced houses). The median might be a better representative of typical house prices.

Step5: Analyze statement (e)

The 5 - number summary consists of the minimum, the first quartile ($Q_1$), the median, the third quartile ($Q_3$), and the maximum, not the mean and standard deviation.

Step6: Analyze statement (f)

The mean is not a resistant parameter because it is affected by extreme values (outliers). For example, adding a very large value to a data - set will increase the mean.

Step7: Analyze statement (g)

The standard deviation measures the amount of variation or dispersion of a set of values. If two populations have the same standard deviation, they have the same amount of variation among their data.

Answer:

C. (a) and (f)