Question

1. in minneapolis, the low temperatures in degrees fahrenheit for nine days in february are as follows: -10, 6, 0, -5, -3, -1, 4, 12, and -2. which one of the following statements about this data set is correct? a. the mean must be smaller than the median because the data set includes more negative values than positive values. b. because q1 is equal to 6 and q3 is equal to 8, the iqr is equal to 2. c. changing the highest temperature from a 12 to a -12 will lower both the mean and the median. d. the standard deviation of the data set will be negative because more than half of the data values are negative. e. 25% of the temperatures in this data set are below -5.

Explanation:

Step1: Arrange data in ascending order

$-10,-5,-3,-2,-1,0,4,6,12$

Step2: Calculate mean

Mean $\bar{x}=\frac{-10 + 6+0 - 5-3 - 1+4+12 - 2}{9}=\frac{-9}{9}=-1$

Step3: Find median

Since $n = 9$ (odd), median is the 5 - th value. Median $=-1$

Step4: Analyze option A

Mean $=-1$, median $=-1$, so option A is wrong.

Step5: Calculate quartiles

Lower half of data: $-10,-5,-3,-2,-1$. $Q_1=-3$
Upper half of data: $0,4,6,12$. $Q_3 = 5$
$IQR=Q_3 - Q_1=5-(-3)=8$, so option B is wrong.

Step6: Analyze effect of changing 12 to - 12

Original mean $=-1$. New sum $=-9-12 - 12=-33$. New mean $=\frac{-33}{9}=-\frac{11}{3}\approx - 3.67$
Original median $=-1$. After changing, data in ascending order: $-12,-10,-5,-3,-2,-1,0,4,6$. Median $=-2$. So option C is correct.

Step7: Recall property of standard - deviation

Standard deviation $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}}$, it is non - negative, so option D is wrong.

Step8: Analyze option E

There are 9 data points. $25\%\text{ of }9 = 2.25$. The 2 - nd value in ordered data is $-5$. But $25\%$ of data points means the value between 2 - nd and 3 - rd in ordered data (using interpolation for quartiles), and not all $25\%$ of data is below $-5$, so option E is wrong.

Answer:

C. Changing the highest temperature from a 12 to a -12 will lower both the mean and the median.