To determine what effect the salespeople had on purchases, a department store polled 700 shoppers as to whether or not they had made a purchase and whether or not they were pleased with the service. Of those who had made a purchase, 125 were happy with the service and 111 were not. Of those who had made no purchase, 148 were happy with the service and 316 were not.
Find the probability that a shopper who was happy with the service had made a purchase (round the answer to the nearest hundredth).
What can you conclude?
About what percent of those who were happy with the service made a purchase?
the transition matrix would be
..... : happy not happy
....buy 125/236 111/236
not buy 148/464 316/464 or
│.53 .47 │
│.32 .68 │
so [a b] x [transition matrix] = [a b]
where a+b= 1
I got .53a + .32b = a or 32b = 47a
and .47a + .68b = b
we only need the first of these relations and the fact that a+b = 1
then a = 1-b
and 32b = 47a
32b = 47(1-b)
32b = 47 - 47b
79b = 47
b = .595 then a = .405
(This is the first time I have seen this kind of question in this forum, and its been over 12 years since I have done one like it. So I am not 100% sure of my work here)
mr.ruiz made a mape of his ranch in the griad below what persecnt of the ranch is used for sheep
To find the probability that a shopper who was happy with the service had made a purchase, we need to use the concept of conditional probability.
Let's define the following events:
A = Shopper made a purchase
B = Shopper is happy with the service
We are given:
P(A) = 125/700 (125 out of 700 shoppers made a purchase)
P(A') = 1 - P(A) = 1 - 125/700 = 575/700 (575 out of 700 shoppers did not make a purchase)
P(B|A) = 125/236 (out of the shoppers who made a purchase, 125 were happy)
P(B|A') = 148/891 (out of the shoppers who did not make a purchase, 148 were happy)
Now, applying Bayes' theorem, we have:
P(A|B) = (P(B|A) * P(A)) / (P(B|A) * P(A) + P(B|A') * P(A'))
Substituting the values we have:
P(A|B) = (125/236 * 125/700) / (125/236 * 125/700 + 148/891 * 575/700)
P(A|B) = 0.281
So, the probability that a shopper who was happy with the service had made a purchase is approximately 0.28 (rounded to the nearest hundredth).
From this result, we can conclude that around 28.1% of shoppers who were happy with the service made a purchase.
Please note that the sample size in this study is relatively small, so these results may not be representative of the entire population.
To find the probability that a shopper who was happy with the service had made a purchase, we need to use conditional probability.
The probability of a shopper making a purchase and being happy with the service can be calculated by dividing the number of shoppers who made a purchase and were happy with the service (125) by the total number of shoppers who were happy with the service (125 + 148).
P(Purchase | Happy with service) = Number of shoppers who made a purchase and were happy with the service / Total number of shoppers who were happy with the service
P(Purchase | Happy with service) = 125 / (125 + 148)
P(Purchase | Happy with service) ≈ 0.458
Therefore, the probability that a shopper who was happy with the service had made a purchase is approximately 0.46.
Now, let's analyze what we can conclude from the given information.
If we compare the probability of making a purchase when happy with the service (0.46) to the overall probability of making a purchase (125 / 700 ≈ 0.18), we can see that there is a higher likelihood of making a purchase when happy with the service. This suggests that salespeople have a positive effect on purchases.
About what percent of those who were happy with the service made a purchase can be calculated by multiplying the conditional probability by 100.
Percent of happy shoppers who made a purchase = P(Purchase | Happy with service) * 100
Percent of happy shoppers who made a purchase ≈ 0.46 * 100 ≈ 46%
Therefore, approximately 46% of those who were happy with the service made a purchase.