7.2 Constructing a Confidence Interval for a Population Mean

• What is the first step in constructing a confidence interval for a population mean?

  Identify sample statistics
• The margin of error is calculated by multiplying the critical value by the standard error.


  True
• The sample standard deviation $s$ measures the spread or variability of the sample data
• A larger sample size ensures more accurate inference about the population mean
• Why is the sample size n</latex> important for constructing a confidence interval?

  Ensures accurate inference
• A higher confidence level results in a wider confidence interval.


  True
• The sample mean ($\bar{x}$) is the average value calculated from the sample data
• The sample standard deviation ($s$) is the square root of the sample variance
• Match the sample size with the corresponding confidence interval width:

  30 ↔️ Wider

  50 ↔️ Narrower

  100 ↔️ Narrowest
• A higher confidence level requires a wider confidence interval.


  True
• Common confidence levels and their associated critical values are: 90% with $z =$$1.645$, 95% with $z =$$1.96$, and 99% with $z =$$2.576$.confidence
• If you estimate a population mean with a 95% confidence level, you can be 95% sure that the true mean falls within the calculated interval
• To calculate the margin of error, use the formula: critical value $\times$ standard error
• The sample standard deviation measures the spread of data around the sample mean.

  True
• The standard error of the mean is calculated using the formula \frac{s}{\sqrt{n}}</latex>, where $s$ is the sample standard deviation and $n$ is the sample size
• With a 95% confidence level, you can be 95% sure that the true population mean falls within the calculated interval.


  True
• For a 95% confidence level and 29 degrees of freedom, the critical value is approximately 2.045

  True
• If the critical value is 1.96 and the standard error is 0.5, the margin of error is 0.98


  True
• Steps to construct a 95% confidence interval for a sample with mean 85, standard deviation 6, and sample size 25:

  1️⃣ Calculate the standard error: $\frac{6}{\sqrt{25}} =$$1.2$

  2️⃣ Find the critical value for 95% confidence and 24 degrees of freedom: 2.064</step_end><step_start>Calculate the margin of error: 2.064 × 1.2 = 2.48</step_end><step_start>Construct the confidence interval: $85 \pm 2.48 =$$[82.52, 87.48]$
• The higher the confidence level, the wider the confidence interval will be.
• The standard error of the mean is calculated using the formula: \frac{s}{\sqrt{n}}
• Steps to construct a confidence interval for a population mean

  1️⃣ Identify sample statistics

  2️⃣ Determine the sample size

  3️⃣ Calculate the standard error of the mean

  4️⃣ Choose a confidence level

  5️⃣ Find the critical value

  6️⃣ Calculate the margin of error

  7️⃣ Construct the confidence interval
• What is the purpose of calculating sample statistics when constructing a confidence interval?

  Estimate true population mean
• How is the sample mean \bar{x}</latex> used in constructing a confidence interval?

  Estimates population mean
• The standard error of the mean measures the variability of sample means around the true population mean
• Match the confidence level with its corresponding critical value:

  90% ↔️ 1.645

  95% ↔️ 1.96

  99% ↔️ 2.576
• The sample standard deviation measures the spread of sample data around the mean.


  True
• What is the sample size ($n$)?


  Number of observations in the sample
• The standard error of the mean measures the variability of sample means around the true population mean
• Match the confidence level with its critical value ($z$):


  90% ↔️ 1.645

  95% ↔️ 1.96

  99% ↔️ 2.576
• A higher critical value results in a wider confidence interval.


  True
• Steps to find the critical value for a t-distribution:

  1️⃣ Identify the confidence level needed.

  2️⃣ Determine the degrees of freedom ($df =$$n - 1$) based on the sample size.

  3️⃣ Use a t-table or calculator to find the critical value associated with these parameters.
• What is the formula for standard error?

  SE = \frac{s}{\sqrt{n}}</latex>
• What is the formula for calculating the sample mean?

  $\bar{x} =$$\frac{\sum x_{i}}{n}$
• Match the statistic with its definition and formula:

  Standard Error of the Mean ↔️ Variability of sample means around the population mean, $\frac{s}{\sqrt{n}}$

  Sample Standard Deviation ↔️ Spread of sample values, No formula

  Sample Size ↔️ Number of observations in the sample, No formula
• To find the critical value for a given confidence level and degrees of freedom in a t-distribution, you use a t-table
• Match the confidence level with its typical critical value for large degrees of freedom:

  90% ↔️ 1.645

  95% ↔️ 1.96

  99% ↔️ 2.576
• What is the formula to calculate the margin of error?

  $critical \times standard\ error$
• A 95% confidence interval indicates that the true population mean lies within the calculated range with 95% certainty


  True
• What degrees of freedom are used when finding the critical value for a confidence interval?

  n - 1</latex>