Answer to Question #225416 in Statistics and Probability for qasim naveed

Question

A company manufactured six television sets on a given day, and these TV sets were inspected for being good or defective. The results of the inspection follow. Good ; Good; Defective; Defective; Good ;Good;

a. What proportion of these TV sets are good?

b. How many total samples (without replacement) of size five can be selected from this population? c. List all the possible samples of size five that can be selected from this population and calculate the sample proportion, of television sets that are good for each sample. Prepare the sampling distribution of d. For each sample listed in part c, calculate the sampling error?

Answer 1

(a). $A\:proportion\:of\:good\:tv\:sets=\frac{4}{6}=\frac{2}{3}$ as we have 4 good TVs over a total of 6TVs

(b). From 6 tvs,5 can be selected without replacement in $6C_5$ number of ways.

$6C_5=\frac{fact\left(6\right)}{fact\left(5\right)}\times fact\left(1\right)=6$

So, without replacement,6 samples of size 5 can be drawn from the population.

(c).

$p_{hat}$ that are good for each sample

We prepare the sampling distribution of $p_{hat}$

$The\:number\:of\:all\:possible\:samples=6C_4=15$

The required table is:

Now the probability of $p_{hat}$ is:

$p_{hat}=0.5\:with\:probability\:\frac{6}{15}$

$p_{hat}=0.75\:with\:probability\:\frac{8}{15}$

$p_{hat}=1.0\:with\:probability\:\frac{1}{15}$

(d). For each sample, sampling error is nothing but the absolute value of the difference between actual proportion of good tv sets i.e. 0.66 and the corresponding $p_{hat}$

List of sampling errors;-