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
Ludmilka [50]
3 years ago
9

In a random sample of 150 customers of a high-speed Internetprovider, 63 said that their service had been interrupted one ormore

times in the past month.A) Find a 95% confidence interval for the proportion of customerswhose service was interrupted one or more times in the pastmonth.B) Find a 99% confidence interval for the proportion of customerswhose service was interrupted one or more times in the pastmonth.C) Find the sample size needed for a 95% confidence interval tospecify the proportion to within -+ 0.05.D) Find the sample size needed for a 99% confidence interval tospecify the proportion to within -+ 0.05.
Mathematics
1 answer:
erastovalidia [21]3 years ago
5 0

Answer:

a) The 95% confidence interval would be given by (0.341;0.499)

b) The 99% confidence interval would be given by (0.316;0.524)

c) n=335

d)n=649

Step-by-step explanation:

1) Notation and definitions

X_{IS}=63 number of high speed internet users that had been interrupted one or more times in the past month.

n=150 random sample taken

\hat p_{IS}=\frac{63}{150}=0.42 estimated proportion of high speed internet users that had been interrupted one or more times in the past month.

p_{IS} true population proportion of high speed internet users that had been interrupted one or more times in the past month.

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.

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 population proportion have the following distribution

p \sim N(p,\sqrt{\frac{\hat p(1-\hat p)}{n}})

1) Part a

In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 95% of confidence, our significance level would be given by \alpha=1-0.95=0.05 and \alpha/2 =0.025. And the critical value would be given by:

t_{\alpha/2}=-1.96, t_{1-\alpha/2}=1.96

The confidence interval for the mean is given by the following formula:

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

If we replace the values obtained we got:

0.42 - 1.96\sqrt{\frac{0.42(1-0.42)}{150}}=0.341

0.42 + 1.96\sqrt{\frac{0.42(1-0.42)}{150}}=0.499

The 95% confidence interval would be given by (0.341;0.499)

2) Part b

In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 99% of confidence, our significance level would be given by \alpha=1-0.99=0.01 and \alpha/2 =0.005. And the critical value would be given by:

t_{\alpha/2}=-2.58, t_{1-\alpha/2}=2.58

The confidence interval for the mean is given by the following formula:

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

If we replace the values obtained we got:

0.42 - 2.58\sqrt{\frac{0.42(1-0.42)}{150}}=0.316

0.42 + 2.58\sqrt{\frac{0.42(1-0.42)}{150}}=0.524

The 99% confidence interval would be given by (0.316;0.524)

3) Part c

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 0.05 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)

And replacing into equation (b) the values from part a we got:

n=\frac{0.42(1-0.42)}{(\frac{0.05}{1.96})^2}=374.32

And rounded up we have that n=335

4) Part d

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 0.05 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)

And replacing into equation (b) the values from part a we got:

n=\frac{0.42(1-0.42)}{(\frac{0.05}{2.58})^2}=648.599

And rounded up we have that n=649

You might be interested in
42 square feet to square inches
dolphi86 [110]

Answer:

6048 square inches

Step-by-step explanation:

7 0
3 years ago
Read 2 more answers
Actual
Katena32 [7]

Answer:

Step-by-step explanation:

so first you multiply how much she jogged then put the in infront of the number

3 0
3 years ago
Graph y = - 3x + 7 .
lbvjy [14]

Answer:

put your first point at 7 in y axis that point go down 3 to the rights side 1 and your point should be at (1,4) then connect the dots and that's how you graph it

Step-by-step explanation:

5 0
2 years ago
A car travels 180 miles in 3 hours. At the same rate of speed, how far will the car travel in h hours
Igoryamba

Answer:

60h

Step-by-step explanation:

180/3: Divide 180 by 3 to get the rate of change.

6 0
2 years ago
I don’t know my answer
Angelina_Jolie [31]
That is why you have an education go to school and learn your anwer.
3 0
3 years ago
Other questions:
  • What is the product of (-7)(-4)?
    7·1 answer
  • Which of these descriptions does the equation 11x - 8 = 4 ​represent?
    9·1 answer
  • Jessica has collected bead all
    13·1 answer
  • What is the standard unit of measurement in the metric system
    8·1 answer
  • 4n - 30 = 7(-2n + 2) +10
    10·2 answers
  • A researcher posts a radio advertisement offering $35 in exchange for participation in a short study. The researcher accepts the
    14·1 answer
  • Jillian says that 1 2/3 equals 3/5
    8·1 answer
  • A department of transportation's study on driving speed and miles per gallon for midsize automobiles resulted in the following d
    14·1 answer
  • A stereo normally priced at $399 is on sale for 15% off. Find the discount and the sale price.
    11·1 answer
  • An airplane cruises 1 kilometer in 1/12 of a minute. What is its cruising speed?
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!