A local swim club offers competitive meets throughout the season. Some races are open competitions with no qualifying times. Other races are qualified competitions. For example, 15- to 16-year-old girls are eligible to swim the 100-meter freestyle in a qualified competition if they posted a time of 1-minute 8-second time or less during the qualifying period.

Consider the group of 15- to 16-year-old females who swim the 100-meter freestyle in the open competitions and the group of 15- to 16-year-old females who swim the 100-meter freestyle in the qualified competitions. Which group’s distribution of race times would most likely have the largest standard deviation?

The group competing in the open competitions is most likely to have a higher standard deviation than the group competing in the qualified competitions.

The group competing in the qualified competitions is most likely to have a higher standard deviation than the group competing in the open competitions.

Both groups would likely have the same standard deviation.

There is not enough information to predict the relative sizes of their standard deviations.

The first option is correct.

The group competing in the open competitions is most likely to have a higher standard deviation than the group competing in the qualified competitions.

Step-by-step explanation:

The standard deviation of a dataset measures how the dataset spreads round a particular average value.

For the qualified competitions, there's a standing rule that only 15-16 year olds that post swimming times of 1 minute 8 seconds or less in the 100 m swimming race can participate in this competition. This means, all the participants of this particular competition are good swimmers relatively and the spread of their resulting swimming times should be just around the 1 minute and 8 second. If there will be outliers, it'll be a very few one.

Therefore, the standard deviation of their swimming times will not be so large as all the data focuses around 1 minute and 8 seconds.

But in the open competition, everyone can participate. It is open to skilled, unskilled, fast, slow, shy, brave, basically all types of swimmers. And with this diverse talent pool comes the large variation in swimming times. Plenty unskilled and slow people will post swimming times that are way more than 1 minute & 8 seconds, some few good ones will post numbers that are good enough and close to 1 minute, 8 seconds. At the end of the day, the large variation in swimming times across the talent spectrum leads to a large standard deviation.

So, the group competing in the open competitions is most likely to have a higher standard deviation than the group competing in the qualified competitions.

Only if not too many people participate in eirher of the competitions can this reasoning be faulty.

a negative exponent is translated as follows:

so, we have