Use the t-distribution to find a confidence interval for a mean u given the relevant sample results. Give the best point estimate for u, the margin of error, and the confidence interval. Assume the results come from a random sample from a population that is approximately normally distributed. A 95% confidence interval for u using the sample results I = 93.1, s = 6.7, and n = 42 Round your answer for the point estimate to one decimal place, and your answers for the margin of error and the confidence interval to two decimal places. point estimate = i margin of error =
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![### Finding a Confidence Interval for a Mean
Use the t-distribution to find a confidence interval for a mean \( \mu \) given the relevant sample results. Give the best point estimate for \( \mu \), the margin of error, and the confidence interval. Assume that the results come from a random sample from a population that is approximately normally distributed.
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#### Example Problem
A 95% confidence interval for \( \mu \) using the sample results \( \overline{x} = 93.1 \), \( s = 6.7 \), and \( n = 42 \).
1. Round your answer for the point estimate to one decimal place.
2. Round your answers for the margin of error and the confidence interval to two decimal places.
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**Point Estimate:**
\[ \overline{x} = \]
**Margin of Error:**
\[ \text{Margin of error} = \]
**95% Confidence Interval:**
\[ \text{The 95% confidence interval is} \]
\[ \ \text{to} \ \]
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**Instructions:**
- Fill in the provided text boxes with the appropriate values.
- Use the t-distribution tables or a statistical calculator to find the critical value.
- Calculate the margin of error using the formula:
\[ \text{Margin of Error} = t^* \left( \frac{s}{\sqrt{n}} \right) \]
where \( t^* \) is the critical value from the t-distribution for a given confidence level.
- The confidence interval is then given by:
\[ \overline{x} \pm \text{Margin of Error} \]
Save your progress if needed by clicking "Save for Later," and submit your final answers by clicking "Submit Answer."
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**User Interaction Elements:**
- Three fields are provided for input: point estimate, margin of error, and confidence interval range.
- A "Submit Answer" button is available to finalize the inputs.
- There is also an option to "Save for Later" if the user needs to pause their attempt and return later.
**Attempts Tracking:**
- The number of attempts used out of the allowed attempts is displayed to guide the user.
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**Important Notes:**
- Ensure all answers are entered to the specified decimal places to avoid rounding errors.
- Double-check calculations and interval values before submission.
Happy Learning!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F915c192e-d8ba-4859-b60e-5e514228af12%2F559141fd-2da7-4f2e-8cdb-76a04ae2d328%2Fm8a2k5a_processed.jpeg&w=3840&q=75)
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