Thank you Junior Achievement USA and the Allstate Foundation surveyed teenagers
ID: 3367953 • Letter: T
Question
Thank you Junior Achievement USA and the Allstate Foundation surveyed teenagers aged 14 to 18 and asked at what age they think they will become financially independent. The responses of 944 teenagers who answered this survey question as follows: Age Financially Independent 16 to 20 21 to 24 25 to 27 28 or older of Responses Number 151 487 244 62 Consider the experiment of randomly selecting a teenager from the population of teenagers aged to 14 to 18. a Compute the probability of being financially independent for each of the four age categories. (5 pts) b. What is the probability of being financially independent before the age of 25? (5 pts) c. What is the probability of being financially independent after the age of 24? (5 pts) d. Do the probabilities suggest that teenagers may be somewhat unrealistic in their expectations about when they will become financially independent? (5 pts)
Explanation / Answer
We have to perform the experiment of randomly selecting a teenager from the population of teenagers aged to 14 to 18.
ANSWER TO PART (a)
(i) Probability of being finacially independent for the age category 16 to 20
we can find the desired probability as follows:
P(16 to 20) = total no. of ways of selecting a response from the total responses of age group 16 to 20 (favourable outcome) / total no. of ways of selecting a response from all the age groups (total no. of outcomes)
we know,
total no. of ways of selecting a response from the total responses of age group 16 to 20 = 151C1 = 151 ways
total no. of ways of selecting a response from all the age groups = 944C1 = 944 ways
therfore,
P(16 to 20) = 151/944 = 0.1599 = 0.16 (roounded off)
(ii) Probability of being finacially independent for the age category 21 to 24
we can find the desired probability as follows:
P(21 to 24) = total no. of ways of selecting a response from the total responses of age group 21 to 24 (favourable outcome) / total no. of ways of selecting a response from all the age groups (total no. of outcomes)
we know,
total no. of ways of selecting a response from the total responses of age group 21 to 24 = 487C1 = 487 ways
total no. of ways of selecting a response from all the age groups = 944C1 = 944 ways
therfore,
P(21 to 24) = 487/944 = 0.516 = 0.52 (rounded off)
(iii) Probability of being finacially independent for the age category 25 to 27
we can find the desired probability as follows:
P(25 to 27) = total no. of ways of selecting a response from the total responses of age group 25 to 27 (favourable outcome) / total no. of ways of selecting a response from all the age groups (total no. of outcomes)
we know,
total no. of ways of selecting a response from the total responses of age group 25 to 27 = 244C1 = 244 ways
total no. of ways of selecting a response from all the age groups = 944C1 = 944 ways
therfore,
P(25 to 27) = 244/944 = 0.2585 = 0.26 (rounded off)
(iv) Probability of being finacially independent for the age category 28 or older
we can find the desired probability as follows:
P(28 or older) = total no. of ways of selecting a response from the total responses of age group 28 or older (favourable outcome) / total no. of ways of selecting a response from all the age groups (total no. of outcomes)
we know,
total no. of ways of selecting a response from the total responses of age group 28 or older = 62C1 = 62 ways
total no. of ways of selecting a response from all the age groups = 944C1 = 944 ways
therfore,
P(28 or older) = 62/944 = 0.0657 = 0.066 (rounded off)
ANSWER TO PART (b)
probability of being financially independent before the age of 25
we know we have 2 age groups before the age of 25, which are
16 to 20
21 to 24
since we already know the individual probabilities of being financially independent for both the above two age groups, in order to find the probability of being financially independent before the age of 25 we just need to add the probabilities for the two age groups above i.e., 16 to 20 and 21 to 24.
that is,
P(before 25) = P(16 to 20) + P(21 to 24)
P(16 to 20) = 0.1599 [from part (a) (i)]
P(21 to 24) = 0.516 [from part (a) (ii)]
therefore,
P(before 25) = 0.1599 + 0.516 = 0.6759 = 0.68 (rounded off)
ANSWER TO PART (c)
probability of being financially independent after the age of 24
we know we have 2 age groups after the age of 24, which are
25 to 27
28 or older
since we already know the individual probabilities of being financially independent for both the above two age groups, in order to find the probability of being financially independent after the age of 24 we just need to add the probabilities for the two age groups above i.e., 25 to 27 and 28 or older.
that is,
P(after 24) = P(25 to 27) + P(28 or older)
P(25 to 27) = 0.2585 [from part(a) (iii)]
P(28 or older) = 0.0657 [from part (a) (iv)]
therefore,
P(after 24) = 0.2585 + 0.0657 = 0.3242 = 0.32 (rounded off)
ANSWER TO PART (d)
from part (c) and (d), we observe, about 68% of the teenagers surveyed expect to be finacially independent before the age of 25, which is much larger proportion as compared to the remaining proportion of the teenagers surveyed, that is 32%, who think they will be financially independent after the age of 24.
Recent surveyes have found that most of the teenagers are still financially dependent on their parents or have taken the student loan, etc as majority of them complete their graduation around the age of 24 years. Majority of them become financially independent in their mid 20s.
Hence, we can say that, yes, the probabilities suggest that the teenagers may be unrealistic in their expectations about when they will become financially independent.
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