A simple random sample of size n is drawn. The sample mean, x, is found to be 18.2, and the sample standard deviation, s, is found to be 4.5. Click the icon to view the table of areas under the t-distribution. (b) Construct a 95% confidence interval about u if the sample size, n, is 61. Lower bound: ; Upper bound: (Use ascending order. Round to two decimal places as needed.)
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!
![A simple random sample of size \( n \) is drawn. The sample mean, \( \bar{x} \), is found to be 18.2, and the sample standard deviation, \( s \), is found to be 4.5. Click the icon to view the table of areas under the t-distribution.
**(b)** Construct a 95% confidence interval about \( \mu \) if the sample size, \( n \), is 61.
**Lower bound:** [ ] ; **Upper bound:** [ ]
(Use ascending order. Round to two decimal places as needed.)
---
**Explanation:**
The task is to construct a 95% confidence interval for the population mean \( \mu \) based on a sample of size 61, with a calculated sample mean (\( \bar{x} \)) of 18.2 and a sample standard deviation (\( s \)) of 4.5.
1. **Identify the critical value** from the t-distribution for a 95% confidence level and 60 degrees of freedom (since degrees of freedom = \( n-1 \)).
2. **Calculate the standard error** of the mean: SE = \( \frac{s}{\sqrt{n}} \).
3. **Construct the confidence interval** using the formula:
\[
\bar{x} \pm (t \times \text{SE})
\]
where \( t \) is the critical value obtained from the t-distribution table.
Make sure to round your final answers to two decimal places.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F573ed9f6-d6e0-426a-94e9-10a8d0b6c2c3%2Ffee44eef-3e3a-41bd-99d7-67305865a6ec%2Fz83nf5m_processed.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
