A random sample of 70 observations produced a mean off x =24.1 from a population with a normal distribution and a standard deviation σ=3.68.
Transcribed Image Text:### Confidence Intervals for Population Mean (µ)
#### (a) 90% Confidence Interval
Find a 90% confidence interval for the population mean (µ):
\[ \text{Lower Bound} \leq \mu \leq \text{Upper Bound} \]
#### (b) 95% Confidence Interval
Find a 95% confidence interval for the population mean (µ):
\[ \text{Lower Bound} \leq \mu \leq \text{Upper Bound} \]
#### (c) 99% Confidence Interval
Find a 99% confidence interval for the population mean (µ):
\[ \text{Lower Bound} \leq \mu \leq \text{Upper Bound} \]
### Explanation
This educational content guides you through finding confidence intervals for a population mean (µ) at different confidence levels. The intervals are represented as inequalities, with a blank space for inputs indicating where the calculated lower and upper bounds should be entered. A higher confidence level corresponds to a wider interval, reflecting greater certainty about the inclusiveness of µ within the specified range.
Transcribed Image Text:A random sample of 70 observations produced a mean of \( \bar{x} = 24.1 \) from a population with a normal distribution and a standard deviation \( \sigma = 3.68 \).
Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...
Expert Solution
Step 1: Determining the given information
The sample size is 70, the sample mean is 24.1 and the population standard deviation is 3.68.
![### Confidence Intervals for Population Mean (µ)
#### (a) 90% Confidence Interval
Find a 90% confidence interval for the population mean (µ):
\[ \text{Lower Bound} \leq \mu \leq \text{Upper Bound} \]
#### (b) 95% Confidence Interval
Find a 95% confidence interval for the population mean (µ):
\[ \text{Lower Bound} \leq \mu \leq \text{Upper Bound} \]
#### (c) 99% Confidence Interval
Find a 99% confidence interval for the population mean (µ):
\[ \text{Lower Bound} \leq \mu \leq \text{Upper Bound} \]
### Explanation
This educational content guides you through finding confidence intervals for a population mean (µ) at different confidence levels. The intervals are represented as inequalities, with a blank space for inputs indicating where the calculated lower and upper bounds should be entered. A higher confidence level corresponds to a wider interval, reflecting greater certainty about the inclusiveness of µ within the specified range.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff71a6399-b1b7-4d60-a35b-aee519cd2d65%2F8ab32271-8dc0-4231-926d-59c5ea879ec0%2Fa24q07o_processed.jpeg&w=3840&q=75)
