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grandymaker [24]
3 years ago
7

An automobile manufacturer would like to know what proportion of its customers are not satisfied with the service provided by th

e local dealer. The customer relations department will survey a random sample of customers and compute a 96% confidence interval for the proportion who are not satisfied.(a) Past studies suggest that this proportion will be about 0.19. Find the sample size needed if the margin of the error of the confidence interval is to be about 0.02.(You will need a critical value accurate to at least 4 decimal places.)Sample size:________(b) Using the sample size above, when the sample is actually contacted, 21% of the sample say they are not satisfied. What is the margin of the error of the confidence interval?MoE: ________
Mathematics
1 answer:
monitta3 years ago
3 0

Answer:

a) n =1623

b) ME=2.0538\sqrt{\frac{0.21 (1-0.21)}{1623}}=0.0208    

Step-by-step explanation:

1) Notation and definitions

n random sample taken

\hat p=0.19 estimated proportion of customers are not satisfied with the service provided by the local dealer

Confidence =0.96 or 96%

Me= 0.02 represent the margin of error

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{p(1-p)}{n}})

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 96% of confidence, our significance level would be given by \alpha=1-0.96=0.04 and \alpha/2 =0.02. And the critical value would be given by:

z_{\alpha/2}=-2.0538, z_{1-\alpha/2}=2.0538

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 0.02 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{0.19(1-0.19)}{(\frac{0.02}{2.0538})^2}=1622.912  

And rounded up we have that n=1623

Part b

The margin of error is given by:

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

So then if w replace the value of n obtained from part a we got:

ME=2.0538\sqrt{\frac{0.21 (1-0.21)}{1623}}=0.0208    

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uysha [10]

Answer:

The 90th percentile for the distribution of the total contributions is $6,342,525.

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean \mu and standard deviation \sigma, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean \mu and standard deviation s = \frac{\sigma}{\sqrt{n}}.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For sums of size n, the mean is \mu*n and the standard deviation is s = \sqrt{n}*\sigma

In this question:

n = 2025, \mu = 3125*2025 = 6328125, \sigma = \sqrt{2025}*250 = 11250

The 90th percentile for the distribution of the total contributions

This is X when Z has a pvalue of 0.9. So it is X when Z = 1.28. Then

Z = \frac{X - \mu}{\sigma}

By the Central Limit Theorem

Z = \frac{X - \mu}{s}

1.28 = \frac{X - 6328125}{11250}

X - 6328125 = 1.28*11250

X = 6342525

The 90th percentile for the distribution of the total contributions is $6,342,525.

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2 years ago
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Step-by-step explanation:

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Answer:

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Step-by-step explanation:

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How to put 24.357 in expanded form?
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Josie is picking out wallpaper for a 9 feet by 10 feet accent wall in her new apartment. She has narrowed her choices down to th
elena55 [62]

Answer:

Josie should purchase floral daisy wallpaper.

Step-by-step explanation:

Area = Length × Width

The area of the Josie's accent wall = 9 × 10 = 90 feet²

The cost of Floral Daisy per square feet =\frac{\text{Area of the wallpaper}}{price}

                                                                    = \frac{(1.475\times 9.8)}{14.67}

                                                                    = \frac{14.455}{14.67}

                                                                    = 0.98534 ≈ $0.99/ft²

The cost of total wallpaper for the wall = 90 × 0.99 = $89.10

The cost of Flower Power per square feet = \frac{(8.33\times 12)}{99}

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                                                                      = 1.0096 ≈ $1.01/ft²

The cost of total wallpaper for the wall = 90 × 1.01 = $90.90

total savings = 90.90 - 89.10 = $1.80

∵ The floral daisy wallpaper comes in size = 1.45 ft × 9.8 ft

∴ Josie needs the number of rolls of floral daisy wallpaper to cover her wall = 7

The cost of 7 rolls = 14.67 × 7 = $102.69

To cover her wall with flower power wallpaper, she needs the number of rolls = 2

The cost of 2 rolls = 99 × 2 = $198

We can see the cost of floral daisy wallpaper is less than flower power.

Therefore, Josie should purchase floral daisy wallpaper.

8 0
3 years ago
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