![drewje12](/avatars/42747.jpg)
drewje12
22.06.2020 •
Mathematics
In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. Do you take the free samples offered in supermarkets? About 56% of all customers will take free samples. Furthermore, of those who take the free samples, about 39% will buy what they have sampled. Suppose you set up a counter in a supermarket offering free samples of a new product. The day you were offering free samples, 303 customers passed by your counter. (Round your answers to four decimal places.) (a) What is the probability that more than 180 will take your free sample? (b) What is the probability that fewer than 200 will take your free sample? (c) What is the probability that a customer will take a free sample and buy the product? Hint: Use the multiplication rule for dependent events. Notice that we are given the conditional probability P(buy|sample) = 0.39, while P(sample) = 0.56. (d) What is the probability that between 60 and 80 customers will take the free sample and buy the product? Hint: Use the probability of success calculated in part (c).
Solved
Show answers
More tips
- H Health and Medicine Kinesiology: What is it and How Does it Work?...
- O Other How to Choose the Best Answer to Your Question on The Grand Question ?...
- L Leisure and Entertainment History of International Women s Day: When Did the Celebration of March 8th Begin?...
- S Style and Beauty Intimate Haircut: The Reasons, Popularity, and Risks...
- A Art and Culture When Will Eurovision 2011 Take Place?...
- S Style and Beauty How to Choose the Perfect Hair Straightener?...
- F Family and Home Why Having Pets at Home is Good for Your Health...
- H Health and Medicine How to perform artificial respiration?...
- H Health and Medicine 10 Tips for Avoiding Vitamin Deficiency...
- F Food and Cooking How to Properly Cook Buckwheat?...
Answers on questions: Mathematics
- M Mathematics If sin(theta) 0 and cos(theta) 0 then the terminal point determined by theta is a) quadrant 1. b) quadrant 2. c) quadrant 3. d) quadrant 4...
- M Mathematics Given the following figure, what is the value of x? a. 10 b. 11 c. 92 d. 144...
- M Mathematics find the area of the regular hexagon. a) 36 cm2 b) 64 cm2 c) 96 cm2 d) 345 cm2...
- M Mathematics Me with 9 10 and 11 you can do any you like you...
- M Mathematics Factor completely, then place the factors in the proper location on the grid. 16ax + 4x 2 +16a 2...
- M Mathematics Answer only number seven. also, tell me the final answer and the process...
- M Mathematics What js the value of the expression 10 -16÷2...
- M Mathematics What is the subset of rational numbers...
- M Mathematics Please help. Write out how you solved it....
- M Mathematics A town kept a record, shown below, of the average hourly wage earned by its residents. The wages were recorded in constant 1982 dollars. If the rate of change experienced...
Ответ:
(a) The probability that more than 180 will take your free sample is 0.1056.
(b) The probability that fewer than 200 will take your free sample is 0.9997.
(c) The probability that a customer will take a free sample and buy the product is 0.2184.
(d) The probability that between 60 and 80 customers will take the free sample and buy the product is 0.8005.
Step-by-step explanation:
We are given that about 56% of all customers will take free samples. Furthermore, of those who take the free samples, about 39% will buy what they have sampled.
The day you were offering free samples, 303 customers passed by your counter.
Firstly, we will check that it is appropriate to use the normal approximation to the binomial, that is;
Is np > 5 and n(1-p) > 5
In our question, n = sample of customers = 303
p = probability that customers will take free sample = 56%
So, np =
= 169.68 > 5
n(1-p) =
= 133.32 > 5
Since, both conditions are satisfied so it is appropriate to use the normal approximation to the binomial.
Now, mean of the normal distribution is given by;
Mean,
=
= 169.68
Also, the standard deviation of the normal distribution is given by;
Standard deviation,
= ![\sqrt{n \times p \times (1-p)}](/tpl/images/0691/7125/731a3.png)
=
= 8.64
Let X = Number of people who will take your free sample
The z score probability distribution for normal distribution is given by;
Z =
~ N(0,1)
(a) The probability that more than 180 will take your free sample is given by = P(X > 180) = P(X > 180.5) {Using continuity correction}
P(X > 180.5) = P(
>
) = P(Z > 1.25) = 1 - P(Z < 1.25)
= 1 - 0.8944 = 0.1056
(b) The probability that fewer than 200 will take your free sample is given by = P(X < 200) = P(X < 199.5) {Using continuity correction}
P(X < 199.5) = P(
<
) = P(Z < 3.45) = 0.9997
(c) We are given in the question that of those who take the free samples, about 39% will buy what they have sampled, this means that we have;
P(Buy the product / taken a free sample) = 0.39
So, Probability(customer will take a free sample and buy the product) = P(customer take a free sample)
P(Buy the product / taken a free sample)
= 0.56
0.39 = 0.2184
(d) Now our mean and standard deviation will get changed because the probability of success now is p = 0.2184 but n is same as 303.
So, Mean,
=
=
= 66.18
Standard deviation,
= ![\sqrt{n \times p \times (1-p)}](/tpl/images/0691/7125/731a3.png)
=
= 7.192
Now, the probability that between 60 and 80 customers will take the free sample and buy the product is given by = P(60 < X < 80) = P(59.5 < X < 80.5) {Using continuity correction}
P(59.5 < X < 80.5) = P(X < 80.5) - P(X
59.5)
P(X < 80.5) = P(
<
) = P(Z < 1.99) = 0.9767
P(X
59.5) = P( ![\frac{X-\mu}{\sigma}](/tpl/images/0691/7125/4db78.png)
![\leq](/tpl/images/0691/7125/46cd6.png)
) = P(Z
-0.93) = 1 - P(Z < 0.93)
= 1 - 0.8238 = 0.1762
Therefore, P(59.5 < X < 80.5) = 0.9767- 0.1762 = 0.8005.
Ответ:
128
Step-by-step explanation:
128 below sea level is the same as -128. So 128 is the opposite