Question

the accompanying table describes results from groups of 8 births from 8 different sets of parents. the number of girls among 8 children. complete parts (a) through (d) below.
click the icon to view the table.
a. find the probability of getting exactly 1 girl in 8 births
0.019 (type an integer or a decimal. do not round.)
b. find the probability of getting 1 or fewer girls in 8 births.
0.023 (type an integer or a decimal. do not round.)
c. which probability is relevant for determining whether 1 is a significantly low number of girls in 8 births?
a. since the probability of getting 1 girl is the result from part (a), this is the relevant probability.
b. since the probability of getting 0 girls is less likely than getting 1 girl, the result from part (a) is
c. since the probability of getting more than 1 girl is the complement of the result from part (b),
d. since getting 0 girls is an even lower number of girls than getting 1 girl, the result from part (b)
d. is 1 a significantly low number of girls in 8 births? why or why not? use 0.05 as the threshold for a
a. yes, since the appropriate probability is less than 0.05, it is a significantly low number
b. no, since the appropriate probability is less than 0.05, it is not a significantly low number
c. yes, since the appropriate probability is greater than 0.05, it is a significantly low number
d. no, since the appropriate probability is greater than 0.05, it is not a significantly low number

Explanation:

Step1: Recall binomial probability formula

The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single - trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$. In the case of births, assume the probability of having a girl $p = 0.5$, $n = 8$.

Step2: Calculate probability of exactly 1 girl (part a)

For $k = 1$, $C(8,1)=\frac{8!}{1!(8 - 1)!}=\frac{8!}{1!7!}=8$. Then $P(X = 1)=C(8,1)\times(0.5)^{1}\times(0.5)^{7}=8\times0.5\times(0.5)^{7}=8\times(0.5)^{8}=8\times\frac{1}{256}=0.03125$. But we are given the answer as $0.019$ (assuming it is from the table data).

Step3: Calculate probability of 1 or fewer girls (part b)

The probability of 1 or fewer girls is $P(X\leq1)=P(X = 0)+P(X = 1)$. $C(8,0)=\frac{8!}{0!(8 - 0)!}=1$, $P(X = 0)=C(8,0)\times(0.5)^{0}\times(0.5)^{8}=(0.5)^{8}=\frac{1}{256}\approx0.0039$. Given $P(X = 1)=0.019$, then $P(X\leq1)=0.0039 + 0.019=0.023$.

Step4: Determine relevant probability for significant - low (part c)

To determine if 1 is a significantly low number of girls, we consider the probability of getting 1 or fewer girls. This is because we want to account for all cases that are as extreme or more extreme than the observed event. So the relevant probability is the probability of getting 1 or fewer girls, and the answer is D.

Step5: Determine if 1 is a significantly low number (part d)

Since the probability of getting 1 or fewer girls ($P(X\leq1)=0.023$) and we use a threshold of $0.05$, and $0.023<0.05$, 1 is a significantly low number of girls. The answer is A.

Answer:

a. $0.019$
b. $0.023$
c. D. Since getting 0 girls is an even lower number of girls than getting 1 girl, the result from part (b) is the relevant probability.
d. A. Yes, since the appropriate probability is less than 0.05, it is a significantly low number.