Chapter PI, Problem 31RE

(a)

To determine

To find out how many cars were measured.

Answer to Problem 31RE

  $\text{38}$ cars.

Explanation of Solution

In the question it is given that one measure of the size of an automobile engine is its displacement, total volume of its cylinders. The summary statistics for several models of new cars is also given. Thus, we can say from the summary statistics that $\text{38}$ cars were measured for the study made.

(b)

To determine

To explain why might the mean be so much larger than the median.

Explanation of Solution

In the question it is given that one measure of the size of an automobile engine is its displacement, total volume of its cylinders. The summary statistics for several models of new cars is also given. Thus, we can say from the summary statistics that $\text{38}$ cars were measured for the study made. Thus, we can say that the mean be so much larger than the median because possibly the distribution is skewed to the right so the mean is larger than the median in that case.

(c)

To determine

To describe the centre and the spread of this distribution with appropriate statistics.

Explanation of Solution

In the question it is given that one measure of the size of an automobile engine is its displacement, total volume of its cylinders. The summary statistics for several models of new cars is also given. Thus, we can say from the summary statistics that $\text{38}$ cars were measured for the study made. Thus, we can say that the mean be so much larger than the median. Thus, the centre is the median which is $\text{148}\text{.5}$ cubic inches. And the spread is the interquartile range and that is $\text{126}$ cubic inches.

(d)

To determine

To explain is that engine unusually large.

Answer to Problem 31RE

No, it is not.

Explanation of Solution

In the question it is given that one measure of the size of an automobile engine is its displacement, total volume of its cylinders. The summary statistics for several models of new cars is also given. Thus, we can say from the summary statistics that $\text{38}$ cars were measured for the study made. Thus, we can say that the mean be so much larger than the median. Thus, the centre is the median which is $\text{148}\text{.5}$ cubic inches. And the spread is the interquartile range and that is $\text{126}$ cubic inches. Therefore, we can say that the engine is not very unusually large because it is bigger than the average but smaller than more than $\text{25\%}$ of cars. The upper quartile is at $\text{231}$ cubic inches.

(e)

To determine

To explain are there any engines in this data set that you would consider to be outlier.

Answer to Problem 31RE

No, there are not any engines in this data set that you would consider to be outlier.

Explanation of Solution

In the question it is given that one measure of the size of an automobile engine is its displacement, total volume of its cylinders. The summary statistics for several models of new cars is also given. Thus, we can say from the summary statistics that $\text{38}$ cars were measured for the study made. Thus, we can say that the mean be so much larger than the median. Thus, the centre is the median which is $\text{148}\text{.5}$ cubic inches. And the spread is the interquartile range and that is $\text{126}$ cubic inches. Therefore, there are not any engines in this data set that you would consider to be outlier because $\text{1}\text{.5 IQR is 189 and 105}-189$ is negative so there cannot be any low outliers. And $231+189=420$ so, there are not any cars with engines bigger than this since the maximum has to be at most $105+275=380$ .

(f)

To determine

To explain is it reasonable to expect that about $68\%$ of car engines measure between $\text{88 and 266}$ cubic inches.

Answer to Problem 31RE

No, it is not reasonable to expect that about $68\%$ of car engines measure between $\text{88 and 266}$ cubic inches.

Explanation of Solution

In the question it is given that one measure of the size of an automobile engine is its displacement, total volume of its cylinders. The summary statistics for several models of new cars is also given. Thus, we can say from the summary statistics that $\text{38}$ cars were measured for the study made. Thus, we can say that the mean be so much larger than the median. Thus, the centre is the median which is $\text{148}\text{.5}$ cubic inches. And the spread is the interquartile range and that is $\text{126}$ cubic inches. Therefore, we can say that it is not reasonable to expect that about $68\%$ of car engines measure between $\text{88 and 266}$ cubic inches because the distribution is skewed to the right and this is probably not a good approximation.

(g)

To determine

To explain how would such a conversion affect each of the summary statistics.

Explanation of Solution

In the question it is given that one measure of the size of an automobile engine is its displacement, total volume of its cylinders. The summary statistics for several models of new cars is also given. Thus, we can say that the mean be so much larger than the median. Thus, the centre is the median which is $\text{148}\text{.5}$ cubic inches. And the spread is the interquartile range and that is $\text{126}$ cubic inches. Thus, such a conversion would affect each of the summary statistics because mean, median, range, quartiles, IQR and standard deviation all get multiplied by $16.4$ .