1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
umka21 [38]
3 years ago
7

MNM Corporation gives each of its 500 employees an aptitude test. The scores on the test are normally distributed with a mean of

75 and a standard deviation of 15. A simple random sample of 36 is taken from the population of 500 employees. What is the probability that the average aptitude test score in the sample will be less than 78.69? A new brand of breakfast cereal is being market tested. One hundred boxes of the cereal were given to consumers to try. The consumers were asked whether they liked or disliked the cereal. Their responses are provided below. Construct a 95% confidence interval for the true proportion of people who will like the cereal. Interpret the practical meaning of this interval estimate, in plain English. Assume that nothing is known about the population proportion. With a .95 probability, how large of a sample needs to be taken to provide a margin of error of 9% or less?
Mathematics
1 answer:
djyliett [7]3 years ago
5 0

Answer:

Q1: P(\bar X

Q2: 0.6 - 1.96 \sqrt{\frac{0.6(1-0.6)}{100}}=0.504  

0.6 + 1.96 \sqrt{\frac{0.6(1-0.6)}{100}}=0.696  

And the 95% confidence interval would be given (0.504;0.696).

We are confident 95% that about 50.4% to 69.6% of people would like the cereal

Q3: n=\frac{0.5(1-0.5)}{(\frac{0.09}{1.96})^2}=118.57  

And rounded up we have that n=119  

Step-by-step explanation:

Q1 :MNM Corporation gives each of its 500 employees an aptitude test. The scores on the test are normally distributed with a mean of 75 and a standard deviation of 15. A simple random sample of 36 is taken from the population of 500 employees. What is the probability that the average aptitude test score in the sample will be less than 78.69?

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Let X the random variable that represent the scores of a population, and for this case we know the distribution for X is given by:

X \sim N(75,15)  

Where \mu=75 and \sigma=15

The distribution for the sample mean \bar X is given by:

\bar X \sim N(\mu=75, \frac{\sigma}{\sqrt{n}}=\frac{15}{\sqrt{36}}=2.5)

For this case we want this probability:

P(\bar X

And we can use the z score given by:

z= \frac{\bar X -\mu}{\sigma_{\bar X}}

P(\bar X

Q2: A new brand of breakfast cereal is being market tested. One hundred boxes of the cereal were given to consumers to try. The consumers were asked whether they liked or disliked the cereal. Their responses are provided below.

Liked =60, Disliked =40 , Total =100

Construct a 95% confidence interval for the true proportion of people who will like the cereal. Interpret the practical meaning of this interval estimate, in plain English.

n=100 represent the random sample taken      

X=60 represent the people liked with the cereal

\hat p=\frac{60}{100}=0.6 estimated proportion of people who liked the cereal

\alpha=0.05 represent the significance level (no given, but is assumed)      

Confidence =95% or 0.95  

p= population proportion of people drivers claimed they always buckle up

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

The confidence interval would be given by this formula  

\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}  

For the 95% confidence interval the value of \alpha=1-0.95=0.05 and \alpha/2=0.025, with that value we can find the quantile required for the interval in the normal standard distribution.  

z_{\alpha/2}=1.96  

And replacing into the confidence interval formula we got:  

0.6 - 1.96 \sqrt{\frac{0.6(1-0.6)}{100}}=0.504  

0.6 + 1.96 \sqrt{\frac{0.6(1-0.6)}{100}}=0.696  

And the 95% confidence interval would be given (0.504;0.696).

We are confident 95% that about 50.4% to 69.6% of people would like the cereal

Q3: Assume that nothing is known about the population proportion. With a .95 probability, how large of a sample needs to be taken to provide a margin of error of 9% or less?

The margin of error for the proportion interval is given by this formula:  

ME=z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}} (a)  

And on this case we have that ME =\pm 0.09 and we are interested in order to find the value of n, if we solve n from equation (a) we got:  

n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2} (b)

We can assume that the estimated proportion is 0.5 since we don't have other info provided to assume a different value. And replacing into equation (b) the values from part a we got:  

n=\frac{0.5(1-0.5)}{(\frac{0.09}{1.96})^2}=118.57  

And rounded up we have that n=119  

You might be interested in
PLEASE HELP!
Westkost [7]
Hello!

We know that the rate of change is just like speed as in the last Q so....

usually the rate of change is y/x so in this case, The Correct Answer is 100%

Option B "25".

I Hope my answer has come to your Help. Thank you for posting your question here in Brainly. We hope to answer more of your questions and inquiries soon. Have a nice day ahead! :)
5 0
3 years ago
Read 2 more answers
What is the difference between dash and hyphen
UNO [17]
Hyphens and Dashes. A hyphen joins two or more words together while a dash separates words into parenthetical statements. The two are sometimes confused because they look so similar, but their usage is different. Hyphens are not separated by spaces, while a dash has a space on either side.
8 0
3 years ago
Betty likes to skate at a skate park that is due south of her school and due west of her
AURORKA [14]
The answer will be 12+13 because the skatepark is 12km from school and her favorite game store was 13 km so it’s 12+13=25km
6 0
3 years ago
Ellen Newman has an income of $18,150 and claims 5 exemptions. The amount allowed for each exemption is $1,000. Her state tax is
Vanyuwa [196]
State tax is computed by first subtracting the amount of exemptions (5*$1000=$5000) from annual income and then multiplying it by the state tax.

($18150-$5000)*5%= $657.50
3 0
3 years ago
Maya deposits $5000 into a checking account that pays 0.75% annual interest compounded monthly. What will be the balance after 8
Nana76 [90]

~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill &\$5000\\ r=rate\to 0.75\%\to \frac{0.75}{100}\dotfill &0.0075\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{monthly, thus twelve} \end{array}\dotfill &12\\ t=years\dotfill &8 \end{cases} \\\\\\ A=5000\left(1+\frac{0.0075}{12}\right)^{12\cdot 8}\implies A=5000(1.000625)^{96}\implies A\approx 5309.08

5 0
2 years ago
Other questions:
  • An experiment was conducted to compare the wearing qualities of three types of paint (A, B, and C) when subjected to the abrasiv
    13·1 answer
  • What is 1 and 2/3 divided by 4/5
    10·1 answer
  • Jaylin's test grades in History class are: 92, 89, 85, 89, and 90. The semester final will count as IWULUI.
    8·1 answer
  • NEED HELP!!!!!!!!!!!!!!!!!!!!!
    14·1 answer
  • Is it zero or u defined?
    15·1 answer
  • Can anyone help me please
    12·1 answer
  • How many values are within the limits 100 and 150
    12·1 answer
  • What is the length of arc S shown below?<br> Enter an exact expression.<br> units
    5·2 answers
  • HURRYY PLEASE I RLLY NEED THIS AND I DONT OPEN LINKS SO DONT THINK ABT IT
    12·1 answer
  • What is the difference between-2 and -8
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!