Question

below are iq scores from 30 randomly selected adults. { 78, 82, 88, 88, 89, 93, 97, 98, 98, 98, 100, 101, 101, 102, 103, 104, 106, 106, 107, 109, 110, 115, 116, 118, 118, 119, 119, 122, 128, 130 }. first, give the mean of the data set. part 2 of 6 next, give the median of the data set. part 3 of 6 now give the mode of the data set. if there is more than one, write them in order, separated by commas. part 4 of 6 finally, give the midrange of the data set. part 5 of 6 given the relationship between the mean and median above, what shape is the distribution likely to be? the distribution will probably be skewed to the right. the distribution will probably be skewed to the left. the distribution will be roughly symmetric. part 6 of 6 suppose the first value in the data set is mistakenly recorded as 0.0. how would this affect the mean? the mean would not change. the mean would get larger. the mean would get smaller. how would this affect the median? the median would get smaller. the median would get larger. the median would not change.

Explanation:

Step1: Recall mean formula

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$, where $n = 30$ and $x_{i}$ are the data - points.
$\sum_{i=1}^{30}x_{i}=78 + 82+88+88+89+93+97+98+98+98+100+101+101+102+103+104+106+106+107+109+110+115+116+118+118+119+119+122+128+130=3120$
$\bar{x}=\frac{3120}{30}=104$

Step2: Recall median formula

For $n = 30$ (an even - numbered data set), the median is the average of the $\frac{n}{2}$th and $(\frac{n}{2}+1)$th ordered data values. The 15th value is 103 and the 16th value is 104. Median $=\frac{103 + 104}{2}=103.5$

Step3: Recall mode formula

The mode is the most frequently occurring value. Here, 98 occurs 3 times, more frequently than any other value.

Step4: Recall mid - range formula

The mid - range is $\frac{\text{Minimum value}+\text{Maximum value}}{2}=\frac{78 + 130}{2}=104$

Step5: Analyze distribution shape

Since the mean ($104$) is greater than the median ($103.5$), the distribution is probably skewed to the right.

Step6: Analyze effect on mean

The original sum is 3120. If the first value 78 is changed to 0, the new sum is $3120-78 + 0=3042$. The new mean is $\frac{3042}{30}=101.4$. So the mean would get smaller.

Step7: Analyze effect on median

The median is the average of the 15th and 16th ordered values. Changing the first value from 78 to 0 does not change the position of the 15th and 16th ordered values. So the median would not change.

Answer:

Part 1: 104
Part 2: 103.5
Part 3: 98
Part 4: 104
Part 5: The distribution will probably be skewed to the right.
Part 6: The mean would get smaller.
Part 7: The median would not change.