Math / Data & Statistics

QuestionAbout 72\% of MCC students believe they can achieve the American dream and about 64\% of Ferris State Universtiy students believe they can achieve the American dream. Construct a $95 \%$ confidence interva for the difference in the proportions of Montcalm Community College students and Ferris State University students who believe they can achieve the American dream.
There were 100 MCC students surveyed and 100 FSU students surveyed. a. With $95 \%$ confidence the difference in the proportions of MCC and FSU students who believe they can achieve the American dream is $\square$ (round to 3 decimal places) and $\square$ (round to decimal places). b. A review of what confidence interval means: If many groups of 100 randomly selected MCC students and 100 randomly selected FSU students were surveyed, then a different confidence interval would be produced from each group. About $\square$ percent of these confidence intervals will contain the true population proportion of the difference in the proportions of MCC students and FSU students who believe they can achieve the American dream about $\square$ percent will not contain the true

Studdy Solution

STEP 1

What is this asking? We want to find out how different the percentage of MCC students who believe in the American dream is from the percentage of FSU students who believe the same, and how confident we are in that difference! Watch out! Don't mix up the percentages of students who believe in the American dream with the confidence level of our result!

STEP 2

1. Calculate the point estimate.
2. Calculate the standard error.
3. Calculate the margin of error.
4. Construct the confidence interval.
5. Interpret the confidence interval.

STEP 3

Let's define $\hat{p}_1$ as the proportion of MCC students who believe they can achieve the American dream, so $\hat{p}_1 = 0.72$ .
Similarly, let $\hat{p}_2$ be the proportion of FSU students with the same belief, so $\hat{p}_2 = 0.64$ .

STEP 4

The point estimate of the difference in proportions is simply the difference between the two sample proportions: $\hat{p}_1 - \hat{p}_2 = 0.72 - 0.64 = \mathbf{0.08}$ .
This tells us that in our samples, $\mathbf{8\%}$ more MCC students than FSU students believe they can achieve the American dream.

STEP 5

The standard error measures the variability of the difference in sample proportions.
The formula is: $SE = \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}$ Where $n_1$ and $n_2$ are the sample sizes for MCC and FSU, respectively.
Both are $\mathbf{100}$ .

STEP 6

Let's plug in our values: $SE = \sqrt{\frac{0.72(1 - 0.72)}{100} + \frac{0.64(1 - 0.64)}{100}} = \sqrt{\frac{0.72 \cdot 0.28}{100} + \frac{0.64 \cdot 0.36}{100}} = \sqrt{\frac{0.2016}{100} + \frac{0.2304}{100}}$ $SE = \sqrt{0.002016 + 0.002304} = \sqrt{0.00432} \approx \mathbf{0.0657}$

STEP 7

For a 95% confidence level, the critical value, often denoted as $z$ , is approximately $\mathbf{1.96}$ .
This value corresponds to the point where 95% of the area under the standard normal distribution lies between -1.96 and 1.96.

STEP 8

The margin of error is calculated as: $ME = z \cdot SE$ Plugging in our values: $ME = 1.96 \cdot 0.0657 \approx \mathbf{0.1288}$

STEP 9

The confidence interval is calculated as: $(\hat{p}_1 - \hat{p}_2) \pm ME$ Using our calculated values: $0.08 \pm 0.1288$ This gives us the interval: $(-0.0488, 0.2088)$ .

STEP 10

We are 95% confident that the true difference in proportions lies between $\mathbf{-0.0488}$ and $\mathbf{0.2088}$ .
This means if we were to repeat this survey many times, 95% of the resulting confidence intervals would contain the true difference in population proportions.

STEP 11

Specifically, about 95% of these confidence intervals will contain the true population proportion difference, and about 5% will not.

STEP 12

a. With 95% confidence the difference in the proportions of MCC and FSU students who believe they can achieve the American dream is $\mathbf{-0.049}$ and $\mathbf{0.209}$ (rounded to 3 decimal places). b. About $\mathbf{95}$ percent of these confidence intervals will contain the true population proportion of the difference, and about $\mathbf{5}$ percent will not.

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Studdy Solution

STEP 1

STEP 2

1. Calculate the point estimate.
2. Calculate the standard error.
3. Calculate the margin of error.
4. Construct the confidence interval.
5. Interpret the confidence interval.

STEP 3

Let's define $\hat{p}_1$ as the proportion of MCC students who believe they can achieve the American dream, so $\hat{p}_1 = 0.72$ .
Similarly, let $\hat{p}_2$ be the proportion of FSU students with the same belief, so $\hat{p}_2 = 0.64$ .

STEP 4

The point estimate of the difference in proportions is simply the difference between the two sample proportions: $\hat{p}_1 - \hat{p}_2 = 0.72 - 0.64 = \mathbf{0.08}$ .
This tells us that in our samples, $\mathbf{8\%}$ more MCC students than FSU students believe they can achieve the American dream.

STEP 5

The standard error measures the variability of the difference in sample proportions.
The formula is: $SE = \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}$ Where $n_1$ and $n_2$ are the sample sizes for MCC and FSU, respectively.
Both are $\mathbf{100}$ .

STEP 6

Let's plug in our values: $SE = \sqrt{\frac{0.72(1 - 0.72)}{100} + \frac{0.64(1 - 0.64)}{100}} = \sqrt{\frac{0.72 \cdot 0.28}{100} + \frac{0.64 \cdot 0.36}{100}} = \sqrt{\frac{0.2016}{100} + \frac{0.2304}{100}}$ $SE = \sqrt{0.002016 + 0.002304} = \sqrt{0.00432} \approx \mathbf{0.0657}$

STEP 7

For a 95% confidence level, the critical value, often denoted as $z$ , is approximately $\mathbf{1.96}$ .
This value corresponds to the point where 95% of the area under the standard normal distribution lies between -1.96 and 1.96.

STEP 8

The margin of error is calculated as: $ME = z \cdot SE$ Plugging in our values: $ME = 1.96 \cdot 0.0657 \approx \mathbf{0.1288}$

STEP 9

The confidence interval is calculated as: $(\hat{p}_1 - \hat{p}_2) \pm ME$ Using our calculated values: $0.08 \pm 0.1288$ This gives us the interval: $(-0.0488, 0.2088)$ .

STEP 10

We are 95% confident that the true difference in proportions lies between $\mathbf{-0.0488}$ and $\mathbf{0.2088}$ .
This means if we were to repeat this survey many times, 95% of the resulting confidence intervals would contain the true difference in population proportions.

STEP 11

Specifically, about 95% of these confidence intervals will contain the true population proportion difference, and about 5% will not.

STEP 12

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Studdy Solution

STEP 1

STEP 2

1. Calculate the point estimate.2. Calculate the standard error.3. Calculate the margin of error.4. Construct the confidence interval.5. Interpret the confidence interval.

STEP 3

Let's **define** p^1\hat{p}_1p^​1​ as the proportion of MCC students who believe they can achieve the American dream, so p^1=0.72\hat{p}_1 = 0.72p^​1​=0.72.Similarly, let p^2\hat{p}_2p^​2​ be the proportion of FSU students with the same belief, so p^2=0.64\hat{p}_2 = 0.64p^​2​=0.64.

STEP 4

STEP 5

STEP 6

STEP 7

For a 95% confidence level, the **critical value**, often denoted as zzz, is approximately 1.96\mathbf{1.96}1.96.This value corresponds to the point where 95% of the area under the standard normal distribution lies between -1.96 and 1.96.

STEP 8

The **margin of error** is calculated as: ME=z⋅SEME = z \cdot SEME=z⋅SE Plugging in our values: ME=1.96⋅0.0657≈0.1288ME = 1.96 \cdot 0.0657 \approx \mathbf{0.1288}ME=1.96⋅0.0657≈0.1288

STEP 9

The **confidence interval** is calculated as: (p^1−p^2)±ME(\hat{p}_1 - \hat{p}_2) \pm ME(p^​1​−p^​2​)±ME Using our calculated values: 0.08±0.12880.08 \pm 0.12880.08±0.1288 This gives us the interval: (−0.0488,0.2088)(-0.0488, 0.2088)(−0.0488,0.2088).

STEP 10

We are 95% confident that the true difference in proportions lies between −0.0488\mathbf{-0.0488}−0.0488 and 0.2088\mathbf{0.2088}0.2088.This means if we were to repeat this survey many times, 95% of the resulting confidence intervals would contain the true difference in population proportions.

STEP 11

Specifically, about **95%** of these confidence intervals will contain the true population proportion difference, and about **5%** will not.

STEP 12

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1. Calculate the point estimate.
2. Calculate the standard error.
3. Calculate the margin of error.
4. Construct the confidence interval.
5. Interpret the confidence interval.

Let's define $\hat{p}_1$ as the proportion of MCC students who believe they can achieve the American dream, so $\hat{p}_1 = 0.72$ .
Similarly, let $\hat{p}_2$ be the proportion of FSU students with the same belief, so $\hat{p}_2 = 0.64$ .

For a 95% confidence level, the critical value, often denoted as $z$ , is approximately $\mathbf{1.96}$ .
This value corresponds to the point where 95% of the area under the standard normal distribution lies between -1.96 and 1.96.

The margin of error is calculated as: $ME = z \cdot SE$ Plugging in our values: $ME = 1.96 \cdot 0.0657 \approx \mathbf{0.1288}$

The confidence interval is calculated as: $(\hat{p}_1 - \hat{p}_2) \pm ME$ Using our calculated values: $0.08 \pm 0.1288$ This gives us the interval: $(-0.0488, 0.2088)$ .

We are 95% confident that the true difference in proportions lies between $\mathbf{-0.0488}$ and $\mathbf{0.2088}$ .
This means if we were to repeat this survey many times, 95% of the resulting confidence intervals would contain the true difference in population proportions.

Specifically, about 95% of these confidence intervals will contain the true population proportion difference, and about 5% will not.