5.3 – Births: Sampling Distribution of Sample Proportion: When three births are

Question

  

5.3 – Births: Sampling Distribution of Sample Proportion: When three births are randomly selected, the sample space for genders is bbb, bbg, bgb, bgg, gbb, gbg, ggb, and ggg. Assume that those eight outcomes are equally likely. Construct a table that describes the sampling distribution of the sample proportion of girls from three births. Does the mean of the sample proportions equal the proportion of girls in three births?

a) To construct the table for sampling distribution of sample proportion of girls from three births, you need to identify the sampling distribution of sample proportion of girls from three births.

b) For computing the sample proportion of girls from three births, you have to find values for all possible sample proportions because each sample is equally to occur, the probability of certain proportion value is the number of occurrences of that proportion value divided by the total number of samples.

c) Here I’ve done the first one for you – Sample bbb (which would be 3 boys, 0 girls):

d) Once your calculations are done, the table representing the sample distribution of girls from three births would be placed in the following format: Sample Proportion of Girls Probability bbb 0 1/8

e) Any repeating values should be combined into another table in ascending order.

f) For the last part read the question carefully, and calculate the information requested.

Sample proportion of girls from three births-number of girls from three births - L- 0 Probability = Number of events occur Total number of possible outcomes

Explanation / Answer

Let X be the number of girls out of 3 births.

X can take the values 0,1,2,3

Then the proportion of girls in 3 births is defined as,

p = X/3

Hence p can take the values 0/3 ,1/3, 2/3 , 3/3 i.e. 0, 1/3, 2/3, 1.

Now, P[p=0] = P[X=0] = P[ no girl in 3births] = 1/8 (favourable case is bbb)

P[p=1/3] = P[X=1] = P[ 1 girl in 3births] = 3/8 (favourable cases are gbb,bgb,bbg)

P[p=2/3] = P[X=2] = P[ 2 girls in 3births] = 3/8 (favourable cases are bgg,gbg,ggb)

P[p=1] = P[X=3] = P[ 3 girls in 3 births] = 1/8 (favourable case is ggg)

Hence the sampling distribution of sample proportion of girls p is given by

Mean of sample proportion of girls

=E[p]

= 0*1/8 + 1/3 * 3/8 + 2/3 * 3/8 + 1 * 1/8

= 1/8 + 2/8 + 1/8

= 4/8

= 1/2

From the table we see that the mean of sample proportion of girls ,here, is not equal to any of the sample proportions.

p 0 1/3 2/3 1 Total f(p)=P[p=p] 1/8 3/8 3/8 1/8 1

Related Questions

Navigate

• Browse (All)
• Browse 5
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.