Please help I don't know how to solve this problem :
Trevor is interested in purchasing the local hardware/sporting goods store in the small town of Dove Creek, Montana. After examining accounting records for the past several years, he found that the store has been grossing over $850 per day about 65% of the business days it is open. Estimate the probability that the store will gross over $850 for the following. (Round your answers to three decimal places.)
(a) at least 3 out of 5 business days
(b) at least 6 out of 10 business days
(c) fewer than 5 out of 10 business days
(d) fewer than 6 out of the next 20 business days
If the outcome described in part (d) actually occurred, might it shake your confidence in the statement p = 0.65? Might it make you suspect that p is less than 0.65? Explain.
Yes. This is unlikely to happen if the true value of p is 0.65.
Yes. This is likely to happen if the true value of p is 0.65.
No. This is unlikely to happen if the true value of p is 0.65.
No. This is likely to happen if the true value of p is 0.65.
(e) more than 17 out of the next 20 business days
If the outcome described in part (e) actually occurred, might you suspect that p is greater than 0.65? Explain.
Yes. This is unlikely to happen if the true value of p is 0.65.
Yes. This is likely to happen if the true value of p is 0.65.
No. This is unlikely to happen if the true value of p is 0.65.
No. This is likely to happen if the true value of p is 0.65.
To solve this problem, we need to use the binomial probability formula. The formula is given by:
P(X = k) = nCk × p^k × (1 - p)^(n - k)
Where:
- P(X = k) is the probability of getting exactly k successes
- nCk is the number of ways to choose k successes from n trials, given by n! / (k! × (n - k)!)
- p is the probability of success on a single trial
- (1 - p) is the probability of failure on a single trial
- n is the total number of trials
Now, let's solve each part of the problem step by step:
(a) At least 3 out of 5 business days:
To find the probability of grossing over $850 at least 3 out of 5 days, we need to calculate the probabilities for 3, 4, and 5 days separately, and then sum them up.
P(at least 3 out of 5 days) = P(3) + P(4) + P(5)
(b) At least 6 out of 10 business days:
Similar to part (a), we need to calculate the probabilities for 6, 7, 8, 9, and 10 days separately, and then sum them up.
P(at least 6 out of 10 days) = P(6) + P(7) + P(8) + P(9) + P(10)
(c) Fewer than 5 out of 10 business days:
To find the probability of grossing over $850 fewer than 5 out of 10 days, we need to calculate the probabilities for 0, 1, 2, 3, and 4 days separately, and then sum them up.
P(fewer than 5 out of 10 days) = P(0) + P(1) + P(2) + P(3) + P(4)
(d) Fewer than 6 out of the next 20 business days:
To find the probability of grossing over $850 fewer than 6 out of the next 20 days, we need to calculate the probabilities for 0, 1, 2, 3, 4, and 5 days separately, and then sum them up.
P(fewer than 6 out of 20 days) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5)
(e) More than 17 out of the next 20 business days:
To find the probability of grossing over $850 more than 17 out of the next 20 days, we need to calculate the probabilities for 18, 19, and 20 days separately, and then sum them up.
P(more than 17 out of 20 days) = P(18) + P(19) + P(20)
Now, for parts (d) and (e), we can assess the likelihood of the outcomes based on whether they are unlikely or likely to happen if the true value of p is 0.65.
(d) If the outcome described in part (d) actually occurred and the store grossed over $850 fewer than 6 out of the next 20 days, it would be unlikely to happen if the true value of p is 0.65. In this case, the observed outcome contradicts our expectation based on the given probability.
(e) If the outcome described in part (e) actually occurred and the store grossed over $850 more than 17 out of the next 20 days, it would be likely to happen if the true value of p is 0.65. In this case, the observed outcome aligns with our expectation based on the given probability.
Based on this analysis, the answers would be:
(d) No. This is unlikely to happen if the true value of p is 0.65.
(e) Yes. This is likely to happen if the true value of p is 0.65.