60% of the students at a college are female. If 13 students at that college are selected random, find the probability that 5 or 6 students will be female
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Do students at various universities differ in how sociable they are? Twenty-five students were randomly selected from each of three universities in a region and were asked to report on the amount of time they spent socializing each day with other students. The result for University X was a mean of 5 hours and an estimated population variance of 2 hours; for University Y, M = 4, S2 = 1.5; and for University Z, M = 6, S2 = 2.5. What should you conclude? Use the .05 level.
To find the probability that 5 or 6 students out of 13 will be female, we need to use the binomial probability formula.
The binomial probability formula is given by:
P(x) = C(n, x) * p^x * q^(n-x)
Where:
- P(x) is the probability of exactly x successes
- n is the total number of trials (in this case, the number of students selected)
- x is the number of successful outcomes (in this case, the number of female students selected)
- p is the probability of a successful outcome (in this case, the probability of selecting a female student)
- q is the probability of a failure (in this case, the probability of selecting a male student)
First, let's calculate the probability of selecting a female student:
p = 60% = 0.60
Since there are no details given about the percentage of male students, we can assume that the remaining students are male. Therefore, the probability of selecting a male student (q) would be:
q = 1 - p = 1 - 0.60 = 0.40
Now, let's calculate the probability of selecting 5 female students:
P(5) = C(13, 5) * (0.60)^5 * (0.40)^(13-5)
Using the combination formula, C(n, x) = n! / (x! * (n-x)!), we can calculate:
C(13,5) = 13! / (5! * (13-5)!) = 1287
Substituting the values into the formula:
P(5) = 1287 * (0.60)^5 * (0.40)^(13-5)
Calculate this value to find the probability of exactly 5 female students.
Similarly, we need to calculate the probability of selecting 6 female students:
P(6) = C(13, 6) * (0.60)^6 * (0.40)^(13-6)
Using the combination formula:
C(13, 6) = 13! / (6! * (13-6)!) = 1716
Substituting the values into the formula:
P(6) = 1716 * (0.60)^6 * (0.40)^(13-6)
Calculate this value to find the probability of exactly 6 female students.
Finally, we add the probabilities of getting exactly 5 female students and exactly 6 female students to find the probability that 5 or 6 students will be female:
P(5 or 6) = P(5) + P(6)