Question

Land's Beginning is a company that sells its merchandise through the mail. It is considering buying a list of addresses from a magazine. The magazine claims that at least 25% of its subscribers have high incomes (they define this to be household income in excess of $100,000). Land's Beginning would like to estimate the proportion of high-income people on the list. Checking income is very difficult and expensive but another company offers this service. Land's Beginning will pay to find incomes for an SRS of people on the magazine's list. They would like the margin of error of the 95% confidence

Answer 1

interval for the proportion to be 0.05 or less. Use the guessed value p = 0.25 to find the required sample size.

Answer 2

well this is easy search it up

Answer 3

Search what up?

Answer 4

@vanoss

Answer 5

@jim_thompson5910

Answer 6

To find the min sample size needed, we use this formula


n = p(1-p)(z/E)^2


where


n = number of people in sample
p = proportion
z = critical value (dependent on the confidence interval)
E = margin of error


In this case,


n = unknown for now
p = 0.25
z = 1.96 (this is found using a calculator or a table)
E = 0.05


So plug all that in to get:



n = p(1-p)(z/E)^2


n = 0.25(1-0.25)(1.96/0.05)^2


n = 0.25(0.75)(1.96/0.05)^2


n = 0.1875(1.96/0.05)^2


n = 0.1875(39.2)^2


n = 0.1875(1536.64)


n = 288.12


n = 289 ... round up to the nearest whole number.


So you need at least 289 people in your SRS

Answer 7

@jim_thompson5910 thank you so much you are a life saver

Answer 8

you're welcome

Answer 9

I have a few more if that's okay @jim_thompson5910

Answer 10

sure that's fine

Answer 11

A software company is planning for an upgrade of their software. You must charge customers $100. Are your customers willing to pay this much? You contact a random sample of 40 customers and find that 11 would pay $100 for the upgrade. If the upgrade is to be profitable, you will need to sell it to more than 20% of your customers. Do the sample data give good evidence that more than 20% are willing to buy?

a. Formulate this problem as a hypothesis test. Give the null and alternative hypotheses.

b. Carry out the significance test. Report the test statistic and the P-value.

c. Should you proceed with plans to produce and market the upgrade?
@jim_thompson5910

Answer 12

how far did you get with this one?

Answer 13

Not far at all tbh, i just have this one to finish an assignment

Answer 14

ok let's do it one part at a time

Answer 15

a. Formulate this problem as a hypothesis test. Give the null and alternative hypotheses.

Answer 16

Let p be the population proportion (of the people willing to buy the software)

So the null hypothesis would be

H0: p <= 0.20


And the alternative hypothesis would be

H1: p > 0.20

Answer 17

b. Carry out the significance test. Report the test statistic and the P-value.

Answer 18

okay awesome
and b?

Answer 19

one sec, still looking up info

Answer 20

ok first off, we need to find the test statistic

so we use this formula

z = (phat - p)/(sqrt(pq/n))

where...

z = test statistic
phat = x/n = 11/40 = 0.275
p = 0.20 (hypothesized proportion)
q = 1-p = 0.8
n = 40

Answer 21

So plugging those values into the formula above gives us


z = (phat - p)/(sqrt(pq/n))


z = (0.275 - 0.2)/( sqrt( ((0.2)(0.8))/(40) ) )


z = 1.18585412256314


So the test statistic is 1.18585412256314

Answer 22

The p value is found by finding the area under the standard normal curve to the right of the test statistic z = 1.18585412256314

So essentially you're computing P(z > 1.18585412256314) which means you'll use a calculator or a table. I recommend a calculator.

I used a calculator to get

P(z > 1.18585412256314) = 0.117839956714519

So the p value is 0.117839956714519

Answer 23

Since this value is larger than the default of alpha = 0.05, this means we fail to reject the null hypothesis

So to answer "c. Should you proceed with plans to produce and market the upgrade?", we would say "no, we failed to reject the null hypothesis which means we must accept the null hypothesis. This means p <= 0.20 and that we won't sell it to more than 20% of the customers" or something like that

Answer 24

ugh awesome thank you a million. this was more than helpful
Thanks again!

Answer 25

I'm glad it helped