problem related to probability?

Question

approximately 30% of all homes in a locality have washing machines. what is the probability that in a sample of 80 homes from that localty fewer than 20 would have washing machines?......kindly and 1 can give the solution of this problem

Answer 1

Use a binomial distribution, and add the probabilities for 0 through 19 successes, given p=0.3 and 80 total trials. You should end up with a probability of 0.135223.

Answer 2

Lithiumdeuteride is correct. However, an approximate result can be obtained with less computation via the normal approximation to the binomial distribution.

We have n = 80, p = 0.3. Then mu = np = 24 and sigma = sqrt(n p (1-p)) = 4.099. We want the probability that x <= 19; with "correction for continuity", this becomes x <= 19.5. So the associated Normal(0,1) probability would be Pr(Z < (19.5-mu)/sigma) = Pr(Z < -1.098) = 0.136. Here Z is a Normal(0,1) random variable and the probability is taken from a table of the cumulative normal distribution (well, actually I cheated and used an Excel worksheet function). This is pretty close to Lithiumdeuteride's more exact value of 0.135.

Answer 3

This clearly calls for the normal approximation to the binomial.
Z = (X -np)/√(p(1-p)); Where X = 20, So,
Z = (20 - 80(0.30))/√(0.30(80)(0.70)) = -4.00/4.099 = -0.976
For α = 0.05(?), the critical Z-value is -1.645
Because -0.976 > -1.645, the null hypothesis of equality cannot be rejected and conclude that it cannot be proved that fewer than 20 homes would have washing machines.
And the probability is 0.165 which is greater than α = 0.05
****Edit*****
OK, I give up! JW (below) is correct. But I stand firm that the text book wants the approximation used in this case. I figured that "fewer than 20" was the same as 20 in a continuous distribution. But apparently, that correction for continuity makes a larger difference than I expected. I usually check my answers with my TI-83, but I didn't this time.

Answer 4

When finding the probability of two things happening together, you multiply the two percentages.

In this case:

30% = 0.30
20/80 = 0.25 (since it is equal or less than 20 out of 80).

0.30*0.25 = 0.075 or 7.5% chance.

Answer 5

100%

Answer 6

whose greater 0.05 or 0.976