Mathematics, 10.04.2020 17:53, fish64

Seams Personal advertises on its website that 95% of customer orders are received within four working days. They performed an audit from a random sample of 500 of the 6,000 orders received that month and it shows 470 orders were received on time. Part A: Can we use a normal approximation? Explain. (5 points) Part B: If Seams Personal customers really receive 95% of their orders within four working days, what is the probability that the proportion in the random sample of 500 orders is the same as the proportion found in the audit sample or less? (5 points) (10 points)

Answers: 3

Show answers

Answers

answered: billy12008

Using the normal distribution and the central limit theorem, it is found that:

a) Since both $np \geq 10$ and $n(1 - p) \geq 10$ , a normal approximation can be used.b) There is a 0.1515 = 15.15% probability that the proportion in the random sample of 500 orders is the same as the proportion found in the audit sample or less.Normal Probability Distribution

In a normal distribution with mean $\mu$ and standard deviation $\sigma$ , the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

It measures how many standard deviations the measure is from the mean. After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X. By the Central Limit Theorem, for a proportion p in a sample of size n, as long as both $np \geq 10$ and $n(1 - p) \geq 10$ , the mean is $\mu = p$ and the standard error is $s = \sqrt{\frac{p(1 - p)}{n}}$

In this problem:

Sample of 500, hence n = 500

.95% of the orders were received on time, hence

Item a:

np = 500(0.95) = 475

n(1 - p) = 500(0.05) = 25

Since both $np \geq 10$ and $n(1 - p) \geq 10$ , a normal approximation can be used.

Item b:

The mean and the standard error are given by:

$\mu = p = 0.95$

$s = \sqrt{\frac{p(1 - p)}{n}} = \sqrt{\frac{0.95(0.05)}{500}} = 0.0097$

The probability that the proportion is of $\frac{470}{500} = 0.94$ or less is the p-value of Z when X = 0.94, hence:

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{0.94 - 0.95}{0.0097}$

Z = -1.03

Z = -1.03 has a p-value of 0.1515.

0.1515 = 15.15% probability that the proportion in the random sample of 500 orders is the same as the proportion found in the audit sample or less.

To learn more about the normal distribution and the central limit theorem, you can take a look at link

answered: maukjackie1974

a. Yes(n=500>=5, n(1-p)=25>=5)

b. 0.15241

Step-by-step explanation:

a. A normal approximation to the binomial can be used $n\geq$ 5 and n(1-p)>=5:

#We calculate our p as follows:

$\hat p$ =x/n=470/500=0.94

n=500

n(1-p)=500(1-0.95)=25

Hence, we can use the normal approximation.

b. This is a normal approximation.

-Given that p=0.95(95%)

-We verify if our distribution can be approximated to a normal:

$np=0.95\times 500=475\\n(1-p)=500(1-0.95)=25\\\\np\geq 5,\ n(1-p)\geq 5$

Hence, we can use the normal approximation of the form:

$P_{bin}(k,n,p)-N(\mu,\sigma^2)\left \{ {{\mu=np=475} \atop {\sigma=\sqrt{np(1-p)}=4.8734}} \right. \\\\\\P_{bin}(k\leq 470)\approx P_{norm}(x\leq 470.5)=P_{norm}(z\leq \frac{470-475}{4.8734})\\\\P_{norm}(z\leq -1.0260)=0.15241$

Hence, the probability of the sample proportion is the same as the proportion of the sample found is 0.15241

answered: Guest

Hope this ; )

answered: Guest

it is c, trust i got your back lol

step-by-step explanation:

Other questions on the subject: Mathematics

Mathematics, 21.06.2019 20:30, shonesam98

Apinhole camera is made out of a light-tight box with a piece of film attached to one side and pinhole on the opposite side. the optimum diameter d(in millimeters) of the pinhole can be modeled by d=1.9[ (5.5 x 10^-4) l) ] ^1/2, where l is the length of the camera box (in millimeters). what is the optimum diameter for a pinhole camera if the camera box has a length of 10 centimeters

Answers: 1

continue

Mathematics, 22.06.2019 03:00, melissadelcid43

Franklin is purchasing a home for $82,107 and is financing 80% of it. the documentary stamp tax on the deed in his state is $0.70 per $100 or portion thereof. how much will he pay for the documentary stamp tax?

Answers: 2

continue

Mathematics, 22.06.2019 03:20, adelheidclees

Why is partitioning a directed line segment into a ratio of 1: 3 not the same as finding the length of the directed line segment?

Answers: 1

continue

Mathematics, 22.06.2019 04:30, daphnevlogs11

Ahmad has completed 24 deliveries so far this week. he needs to make 30 deliveries for the week. what percentage of his deliveries has ahmad completed?

Answers: 1

continue

Do you know the correct answer?

Seams Personal advertises on its website that 95% of customer orders are received within four workin...