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Marina86 [1]
3 years ago
9

In a sample of 1000 randomly, selected consumers who had opportunities to send in a rebate claim form after purchasing a product

, 250 of these people said they never did so. Reasons cited for their behavior included too many steps in the process, amount too small, missed deadline, fear of being placed on a mailing list, lost receipt, and doubts about receiving the money. Calculate a 95% confidence Interval for the true proportion of such consumers who never apply for a rebate.
Mathematics
1 answer:
Alexxx [7]3 years ago
8 0

Answer: =(0.2232,0.2768)

Step-by-step explanation:

The confidence interval for population proportion is given by :-

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

Given : Sample size : n=1000

The proportion of people selected consumers who had opportunities to send in a rebate claim form after purchasing a product said they never did so =\dfrac{250}{1000}=0.25

Significance level : 1-0.95=0.05

Critical value : z_{\alpha/2}=1.96

Now, the 95% confidence Interval for the true proportion of such consumers who never apply for a rebate will be :-

0.25\ \pm (1.96)\sqrt{\dfrac{0.25(1-0.25)}{1000}}\\\\\approx0.25\pm0.0268\\\\=(0.2232,\ 0.2768)

Hence, a confidence Interval for the true proportion of such consumers who never apply for a rebate .

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Step-by-step explanation:

Let us use the first letters of their names to represent them.

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So, from the first statement, we can write out an equation: E = \frac{1}{3}A + 8, that is, <em>one-third of Angela's savings + $8 will give us Eli's savings</em>

From the second statement, we can write out: E + 10 = (A + 10) + 4 which means that <em>if $10 is added to both their savings, Eli will still have $4 more than Angela.</em>

So, we can solve both equations;

Equation 1: E = \frac{1}{3}A + 8

Equation 2: E + 10 = (A + 10) + 4; which can be rewritten as E + 10 = A + 14

and then E = A + 4

The two equations (in bold), can then be solved simultaneously.

Let us carry out this operation to <u>eliminate the E:</u> Subtract equation 1 from equation 2, so that;

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If we check what was said in the question about their savings, which is what we have represented using the equations, the answers can be confirmed.

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