Over the first 20 20 days of the semester, one student is late to class on 6 6 days. Construct a 98 % 98\% confidence interval for the true proportion of time this student is late.

Over the 1st 20 days of the semester, a student is late to class on 6 of those days. We're asked here to construct a 98% confidence interval for the true proportion of time that this student is late. Now if you already worked through the previous practice problem, we actually already calculated the margin of error. But I'm going to go through that process really quickly again here. That was already steps 1 through 4. So really, this problem is just putting that together in step 5 to actually construct our confidence interval. But I'm going to go ahead and go back through that process really quickly going through our steps. We first need to verify that the number of successes and failures is at least 5 because they were late to clay class on 6 days, and that's what we're considering to be a success here. That is at least 5. On the other 14 days, they were late. That is our failures. In this case, that is also at least 5. Moving on to step 2, finding our critical value based on having a 98% confidence interval. That corresponds to a critical z value, z alpha over 2, of 2.326. Then p hat, that's x over n, that was equal to 6 over 20, which is 0.3 dividing x by n. Then we can find our margin of error by taking that z alpha over 22.326, multiplying that by the square root of 0.3 times 1 minus 0.3, so 0.7 over n. That's our total number of days in this case, which is 20 here. I'm just using my margin of error equation, which is right here. Now this worked out to give us a margin of error of 0.238. And from here, we are at step 5 where we're actually constructing our confidence interval. If you wanna go through those first four steps in more details, click back to that previous problem. Alright. Continuing on on step 5 here, we're finding the upper and lower bounds of that confidence interval by subtracting and adding that margin of error to our point estimate, p hot. So our confidence interval here is going to be equal to 0.3 minuteus 0.238 and 0.3 plus 0.238. This results in a confidence interval here. That lower bound is going to give us 0.062 and our upper bound is 0.538. So this is a rather large interval here because our lower bound is really small. Our upper bound is 0.538. So this is our full confidence interval. This is our confidence interval for the true proportion of time that the student is late. Remember, an important part of working with confidence intervals is being able to actually interpret what they mean. Based on what we found here, we can say that we are 98% confident that the true proportion of time that this student is late is between 6.2% and 53.8%. Feel free to let us know if you have any questions, and I'll see you in the next one.

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1. Intro to Stats and Collecting Data1h 14m

Intro to Stats
24m

Levels of Measurement
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Sampling Methods
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Frequency Distributions
35m

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Finding Binomial Probabilities-Excel
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Probabilities & Z-Scores w/ Graphing Calculator
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Finding Probabilities, Z Values, and X Values with the Normal Distribution-Excel
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Sampling Distribution of the Sample Mean and Central Limit Theorem
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Distribution of Sample Mean - Excel
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Introduction to Confidence Intervals
15m

Confidence Intervals for Population Mean
1h 18m

Determining the Minimum Sample Size Required
12m

Finding Probabilities and T Critical Values - Excel
28m

Confidence Intervals for Population Means - Excel
25m

8. Sampling Distributions & Confidence Intervals: Proportion1h 25m

Sampling Distribution of Sample Proportion
29m

Confidence Intervals for Population Proportion
42m

Confidence Intervals for Population Proportion - Excel
12m

9. Hypothesis Testing for One Sample3h 29m

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Hypothesis Testing: Proportions - Excel
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Two Proportions
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Two Proportions Hypothesis Test - Excel
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8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

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8. Sampling Distributions & Confidence Intervals: Proportion

Confidence Intervals for Population Proportion

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Multiple Choice

Over the first $20$ days of the semester, one student is late to class on $6$ days. Construct a $98 percent sign$ confidence interval for the true proportion of time this student is late.

(0.276, 0.324)

(0.131, 0.469)

(0.3, 0.7)

(0.062,0.538)

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Verified step by step guidance

Identify the sample proportion (p̂) by dividing the number of days the student was late (66) by the total number of days (2020). This gives p̂ = 66/2020.

Determine the standard error (SE) of the sample proportion using the formula: SE = sqrt((p̂ * (1 - p̂)) / n), where n is the sample size (2020).

Find the critical value (z*) for a 98% confidence interval. This value corresponds to the z-score that captures the central 98% of the standard normal distribution. You can find this value using a z-table or statistical software.

Calculate the margin of error (ME) by multiplying the critical value (z*) by the standard error (SE): ME = z* * SE.

Construct the confidence interval by adding and subtracting the margin of error from the sample proportion: (p̂ - ME, p̂ + ME). This interval estimates the true proportion of time the student is late with 98% confidence.

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Textbook Question

What type of variable is required to construct a confidence interval for a population proportion?

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Textbook Question

[NW] You Explain It! New Deal Policies In response to the Great Depression, Franklin D. Roosevelt enacted many New Deal policies. One such policy was the enactment of the National Recovery Administration (NRA), which required businesses to agree to wages and prices within their particular industry. The thought was that this would encourage higher wages among the working class, thereby spurring consumption. In a Gallup survey conducted in 1933 of 2025 adult Americans, 55% thought that wages paid to workers in industry were too low. The margin of error was 3 percentage points with 95% confidence. Which of the following represents a reasonable interpretation of the survey results? For those that are not reasonable, explain the flaw.

b. We are 92% to 98% confident 55% of adult Americans during the Great Depression felt wages paid to workers in industry were too low.

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Multiple Choice

Which type of variable is required to construct a confidence interval for a $p$ (population proportion)?

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Multiple Choice

Over the first $20$ days of the semester, one student is late to class on $6$ days. Find the margin of error for a $98 percent sign$ confidence interval for the true proportion of time this student is late.

A previous study found that your school consists of $60\%$ White/Caucasian students. You want the $98\%$ confidence interval for the proportion of White/Caucasian students to be no more than $.05$ away from the true proportion. How many students must you include in a sample to create this confidence interval?

You wish to estimate with $90 percent sign$ confidence the population proportion of people in Gen-Z that use social media. If you want your estimate to be accurate within $4 percent sign$ of the population proportion, what is the minimum sample size needed?

Your company has asked you to estimate the proportion of people who prefer the color red over other primary colors for manufacturing purposes. If they want the estimate to be within $.01$ of the true proportion with $95\%$ confidence, how many people should you survey?

You want to make a $97 percent sign$ confidence interval for the population proportion of people between $20 minus 30$ years old who have gotten a speeding ticket in the past $2$ years. A prior study found that $26 percent sign$ of people between $20 minus 30$ years old have received a speeding ticket in the last year. If you want your estimate to be accurate within $3 percent sign$ of the true population proportion, what is the minimum sample size needed?