Question

.When do creative people get their best ideas? USA Today did a survey of 966 inventors (who hold U.S. patents) and obtained the following information.
Time of Day When Best Ideas Occur
Time Number of Inventors
6 A.M.-12 noon
12 noon-6 P.M.
6 P.M.-12 midnight
12 midnight-6 A.M. 271
123
327
245
(a) Assuming that the time interval includes the left limit and all the times up to but not including the right limit, estimate the probability that an inventor has a best idea during each time interval: from 6 A.M. to 12 noon, from 12 noon to 6 P.M., from 6 P.M. to 12 midnight, from 12 midnight to 6 A.M. (Enter your answers to 3 decimal places.)

6AM-12PM
12PM-6PM

6PM-12AM 12AM-6AM

(b) Do the probabilities add up to 1? Why should they?

No, because they do not cover the entire sample space.Yes, because they cover the entire sample space. Yes, because they do not cover the entire sample space.No, because they cover the entire sample space.


What is the sample space in this problem?

12AM-12PM

6AM-6PM

6AM-12AM

the entire day

Answer 1

Answer:

a. P([6am-12noon)) = 0.272 (or about 27.2%). P([12noon-6pm)) = 0.123 (or about 12.3%). P([6pm-12midnight)) = 0.328 (or about 32.8%). P([midnight-6am)) = 0.246 (or about 24.6%). b. Yes, because they cover the entire sample space. c. The entire day.

Step-by-step explanation:

The questions give us evident insights for the answers.

Sample space

The sample space is all the different possible values for the answers of each inventor and the question asks for the time of day when best ideas occur. The possible options given are four intervals which include the left limit and not the right limit. This gives us all the possible hours of the day inside these intervals:

[6am-12noon); [12noon-6pm); [6pm-12midnight);[midnight-6am).

The symbol "[" indicates that the interval 'takes' the value in the interval, whereas the symbol ")" does not. Thus, the possible answers are the sample space, that is, all possible options.

Probability

Notice also that we have 966 inventors and we have 966 answers (271 + 123 + 327 + 245 = 966). Then, the probabilities are counts for each interval divide by the total answers (996). The answers are rounded to three decimal places.

As a result of all this, we can proceed as follows:

a. Calculation of probabilities

Probability for the interval [6am-12noon)

The number of cases: 271.

Then, the probability is P([6am-12noon)) = 271/996 = 0.272 (or about 27.2%).

Probability for the interval [12noon-6pm)

The number of cases: 123.

Then, the probability is P([12noon-6pm)) = 123/996 = 0.123 (or about 12.3%).

Probability for the interval [6pm-12midnight)

The number of cases: 327.

Then, the probability is P([6pm-12midnight)) = 327/996 = 0.328 (or about 32.8%).

Probability for the interval P([midnight-6am))

The number of cases: 245.

Then, the probability is ([midnight-6am)) = 245/996 = 0.246 (or about 24.6%).

b. Do the probabilities add up to 1? Why should they?

Yes, because they cover the entire sample space.

They should add up to 1 because, as we explained at the beginning, we have all possible answers for a day including in those intervals and the answers for all creative people ask in the survey (996 were asked and the sum for each interval is also 996). Summing up all the probabilities, we should obtain a result approximately equal to 1 (because we round the answers to three decimals). So, 0.272+0.123+0.328+0.246 = 0.969 which is practically 1.

c. What is the sample space in this problem?

The survey asks for the time of the day when the best ideas occur and possible answers should be all hours of the day (sample space). Well, the survey group the data at different intervals that include every possible hour; then, the sample space in this problem is the entire day.