Mathematics, 19.02.2020 01:54 honeytrail429

A 20-minute consumer survey mailed to 400 adults aged 25-34 included a $5 Starbucks gift certificate. The same survey was mailed to 400 adults aged 25-34 without the gift certificate. There were 57 responses from the first group and 31 from the second group.
(a) Perform a two tailed test comparing the response rate (proportions) at a = .05
(b) Form a confidence interval for the difference of proportions, without pooling the sample. Does it includes zero?

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Answer from: ddcope1030

a) $z=\frac{0.1425-0.0775}{\sqrt{0.11(1-0.11)(\frac{1}{400}+\frac{1}{400})}}=2.938$

p_v =2*P(Z2.938)=0.0033

If we compare the p value and the significance level given $\alpha=0.05$ we see that p_v so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the two proportions are different at 5% of significance. .

b) $(0.1425-0.0775) - 1.96 \sqrt{\frac{0.1425(1-0.1425)}{400} +\frac{0.0775(1-0.0775)}{400}}=0.02187$ $(0.1425-0.0775) + 1.96 \sqrt{\frac{0.1425(1-0.1425)}{400} +\frac{0.0775(1-0.0775)}{400}}=0.10813$

And the 95% confidence interval would be given (0.02187;0.10813).

And as we can see the confidence interval not contains the 0.

Step-by-step explanation:

Previous concepts

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".

p_A represent the real population proportion for the gift

$\hat p_A =\frac{57}{400}=0.1425$ represent the estimated proportion for the gift

n_A=400 is the sample size required for the gift

p_B represent the real population proportion for no gift

$\hat p_B =\frac{31}{400}=0.0775$ represent the estimated proportion for no gift

n_B=400 is the sample size required for no gift

represent the critical value for the margin of error

The population proportion have the following distribution

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

Part a

We need to conduct a hypothesis in order to check if the proportions are equal , the system of hypothesis would be:

Null hypothesis: p_A = p_B

Alternative hypothesis: $p_{A} \neq p_{B}$

We need to apply a z test to compare proportions, and the statistic is given by:

$z=\frac{p_{A}-p_{B}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{A}}+\frac{1}{n_{B}})}}$ (1)

Where $\hat p=\frac{X_{A}+X_{B}}{n_{A}+n_{B}}=\frac{51+37}{400+400}=0.11$

Calculate the statistic

Replacing in formula (1) the values obtained we got this:

$z=\frac{0.1425-0.0775}{\sqrt{0.11(1-0.11)(\frac{1}{400}+\frac{1}{400})}}=2.938$

Statistical decision

We have a significance level provided $\alpha=0.05$ , and now we can calculate the p value for this test.

Since is a two tailed test the p value would be:

p_v =2*P(Z2.938)=0.0033

Part b

The confidence interval for the difference of two proportions would be given by this formula

$(\hat p_A -\hat p_B) \pm z_{\alpha/2} \sqrt{\frac{\hat p_A(1-\hat p_A)}{n_A} +\frac{\hat p_B (1-\hat p_B)}{n_B}}$

For the 95% confidence interval the value of and , with that value we can find the quantile required for the interval in the normal standard distribution.

$z_{\alpha/2}=1.96$

And replacing into the confidence interval formula we got: $(0.1425-0.0775) - 1.96 \sqrt{\frac{0.1425(1-0.1425)}{400} +\frac{0.0775(1-0.0775)}{400}}=0.02187$ $(0.1425-0.0775) + 1.96 \sqrt{\frac{0.1425(1-0.1425)}{400} +\frac{0.0775(1-0.0775)}{400}}=0.10813$ And the 95% confidence interval would be given (0.02187;0.10813).

We are confident at 95% that the difference between the two proportions is between $0.02187\leq p_A -p_b \leq 0.10813$

And as we can see the interval not contains the 0.

Answer from: Quest

option: d is the correct answer.

d) none of the above.

step-by-step explanation:

clearly from the figure attached to the question we could see that there is no shape given. hence without any shape given in the figure and after rotation we can't obtained a square pyramid.

since, to obtain a image we need a proper figure of the pre-image so that it can be transformed.

hence, the correct answer is:

d) none of the above.

step-by-step explanation:

Answer from: Quest

30.5*10=350$

you have to multiply the number without the decimal point then add it later but if its tenn u must move the place value of the decimal point 1 to the right.

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