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lapo4ka [179]
3 years ago
12

Direct mail advertisers send solicitations​ ("junk mail") to thousands of potential customers in the hope that some will buy the

​ company's product. The response rate is usually quite low. Suppose a company wants to test the response to a new flyer and sends it to 1030 people randomly selected from their mailing list of over​ 200,000 people. They get orders from 103 of the recipients.
Required:
Create a 95% confidence interval for the percentage of people the company contacts who may buy something. (Show your work. Step by step)
Mathematics
1 answer:
harina [27]3 years ago
5 0

Answer:

The interval is (.0817, .118).

Step-by-step explanation:

Let's make this interval! The easy way is to go to STAT->TEST->A, 1-prop z-int, but I will show the long way.

<u>1. Conditions</u>

First we must check the conditions.

<em>Randomization condition: </em>The sample is given to be random, so we can assume independence.

<em>Success/failure condition: </em>Both np>10 and nq>10 must be met.

  • np = 1030(103/1030) = 103 ✓
  • nq = 1030(927/1030) = 927 ✓

<em>10% condition: </em>The sample cannot be greater than 10% of the population.

  • 10n < N, 10(1030) < 200000 ✓

All conditions are met, so we can use <u>a 1-proportion z-interval</u> to represent the data.

<u></u>

<u>2. Mechanics</u>

<u>Find </u>\hat{p}<u>, the sample proportion.</u>

103/1030 = .1

\hat{q}, then, the probability of not p, is 1 - .1 = .9

<u>Find z* (z star), the critical z-value.</u>

You can do this on your calculator using 2nd->VARS->3 (or you can memorize it - it's 1.96. Winky face.) This is called the invNorm function, that takes the left side area under the normal curve. But what do we plug in to the invNorm function?

invNorm(.975) = 1.96

<u>Plug values into the equation.</u>

\hat{p} ± z*\sqrt{\frac{\hat{p}\hat{q}}{n} }

.1 ± 1.96\sqrt{\frac{.1*.9}{1030} }

.1 ± 1.96(.00935)

.1 - .0183 = .0817

.1 + .0183 = .118

(.0817, .118)

<u>3. Conclusion</u>

Based on this sample, we are 95% confident that the proportion of people the company contacts who may buy something is between .0817 and .118. That isn't very likely, but maybe they'll make a little profit!

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