Podcast
Questions and Answers
Which of the following best defines the confidence interval for a population mean?
Which of the following best defines the confidence interval for a population mean?
- An interval estimate that is likely to contain the population mean (correct)
- An interval containing the sample mean
- An interval that exactly contains the population mean
- A point estimate of the population median
What critical value z_{1-α/2} corresponds to a 95% confidence level?
What critical value z_{1-α/2} corresponds to a 95% confidence level?
- 1.75
- 2.58
- 1.64
- 1.96 (correct)
If the sample size increases, what happens to the width of the confidence interval?
If the sample size increases, what happens to the width of the confidence interval?
- It increases
- It remains the same
- It becomes zero
- It decreases (correct)
In the formula for a confidence interval, what is represented by σ?
In the formula for a confidence interval, what is represented by σ?
What is the value of the standard error if σ = 15 mg/dL and n = 50?
What is the value of the standard error if σ = 15 mg/dL and n = 50?
When constructing a 95% confidence interval for the population mean blood sugar level, what is the margin of error if the sample mean is 120 mg/dL, standard deviation is 15 mg/dL, and sample size is 50?
When constructing a 95% confidence interval for the population mean blood sugar level, what is the margin of error if the sample mean is 120 mg/dL, standard deviation is 15 mg/dL, and sample size is 50?
For which value of α does z_{1-α/2} equal 2.58?
For which value of α does z_{1-α/2} equal 2.58?
What is the interpretation of a 95% confidence interval for the population mean?
What is the interpretation of a 95% confidence interval for the population mean?
Study Notes
Confidence Interval for a Population Mean
- A confidence interval for the population mean μ is constructed using the formula: $X ± z_{1-α/2}(\frac{σ}{\sqrt{n}})$
- The formula involves: sample mean X, critical value z_{1-α/2}, population standard deviation σ, and sample size n
- The critical value z_{1-α/2} is determined by the desired level of confidence and is obtained from the standard normal distribution table or calculated using statistical software
Interpreting Confidence Intervals
- A confidence interval provides an interval estimate for μ with a specified level of confidence
- A 95% confidence interval means that approximately 95% of the intervals constructed using this method would contain the true population mean μ
Example: Estimating Average Blood Sugar Level
- Sample mean X = 120 mg/dL, population standard deviation σ = 15 mg/dL, and sample size n = 50
- To construct a 95% confidence interval, the critical value z_{1-α/2} = 1.96 is used
- The 95% confidence interval is calculated as: $120 ± 1.96(\frac{15}{\sqrt{50}})$
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Description
Learn how to construct a confidence interval for the population mean using the sample mean, standard deviation, and sample size. Understand the formula and its components.
