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When \(\sigma\) is unknown and the sample is of size \(n \geq 30\), there are two methods for computing confidence intervals for \(\mu\). Method 1: Use the Student's \(t\) distribution with \(d . f .=n-1 .\) This is the method used in the text. It is widely employed in statistical studies. Also, most statistical software packages use this method. Method 2: When \(n \geq 30\), use the sample standard deviation \(s\) as an estimate for \(\sigma\), and then use the standard normal distribution. This method is based on the fact that for large samples, \(s\) is a fairly good approximation for \(\sigma .\) Also, for large \(n\), the critical values for the Student's \(t\) distribution approach those of the standard normal distribution. Consider a random sample of size \(n=31\), with sample mean \(\bar{x}=45.2\) and sample standard deviation \(s=5.3\). (a) Compute \(90 \%, 95 \%\), and \(99 \%\) confidence intervals for \(\mu\) using Method 1 with a Student's \(t\) distribution. Round endpoints to two digits after the decimal. (b) Compute \(90 \%, 95 \%\), and \(99 \%\) confidence intervals for \(\mu\) using Method 2 with the standard normal distribution. Use \(s\) as an estimate for \(\sigma\). Round endpoints to two digits after the decimal. (c) Compare intervals for the two methods. Would you say that confidence intervals using a Student's \(t\) distribution are more conservative in the sense that they tend to be longer than intervals based on the standard normal distribution? (d) Repeat parts (a) through (c) for a sample of size \(n=81\). With increased sample size, do the two methods give respective confidence intervals that are more similar?

Step by step solution

01

Determine Critical Values for Method 1 (n=31)

For Method 1, use the Student's t-distribution with degrees of freedom (df) = n - 1 = 31 - 1 = 30. Look up critical t-values for 90%, 95%, and 99% confidence levels in a t-table. - 90% confidence level: approximately 1.697 - 95% confidence level: approximately 2.045 - 99% confidence level: approximately 2.750.

02

Calculate Confidence Intervals Using Method 1 (n=31)

Use the formula for confidence interval: \[ \bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}} \]- For 90%: \( 45.2 \pm 1.697 \cdot \frac{5.3}{\sqrt{31}} \approx 45.2 \pm 1.62, \) thus (43.58, 46.82).- For 95%: \( 45.2 \pm 2.045 \cdot \frac{5.3}{\sqrt{31}} \approx 45.2 \pm 1.94, \) thus (43.26, 47.14).- For 99%: \( 45.2 \pm 2.750 \cdot \frac{5.3}{\sqrt{31}} \approx 45.2 \pm 2.61, \) thus (42.59, 47.81).

03

Determine Critical Values for Method 2 (n=31)

For Method 2, use the standard normal distribution critical values (z-values). - 90% confidence level: approximately 1.645 - 95% confidence level: approximately 1.960 - 99% confidence level: approximately 2.576.

04

Calculate Confidence Intervals Using Method 2 (n=31)

Use the formula for confidence interval: \[ \bar{x} \pm z_{\alpha/2} \cdot \frac{s}{\sqrt{n}} \]- For 90%: \( 45.2 \pm 1.645 \cdot \frac{5.3}{\sqrt{31}} \approx 45.2 \pm 1.57, \) thus (43.63, 46.77).- For 95%: \( 45.2 \pm 1.960 \cdot \frac{5.3}{\sqrt{31}} \approx 45.2 \pm 1.87, \) thus (43.33, 47.07).- For 99%: \( 45.2 \pm 2.576 \cdot \frac{5.3}{\sqrt{31}} \approx 45.2 \pm 2.46, \) thus (42.74, 47.66).

05

Compare Intervals for Methods 1 and 2 (n=31)

Compare the confidence intervals from Steps 2 and 4: - Method 1 intervals are slightly wider than Method 2 intervals, showing that the t-distribution tends to be more conservative for smaller sample sizes.

06

Calculate Confidence Intervals for n=81 Using Method 1

For n=81 (df = 80), the critical t-values are very close to those of the standard normal distribution: - Calculate using similar steps as Step 2 with df = 80. - Confidence Intervals should be computed similarly by looking up the t-values for df = 80 in the t-table, approximating very closely as: - 90%: - 95%: - 99%: Close to z-values due to larger sample size.

07

Calculate Confidence Intervals for n=81 Using Method 2

For Method 2 with n=81: - Use standard normal z-values as in Step 3. - Since n = 81 is large, intervals from this method closely approximate those from Method 1.

08

Analyze Impact of Sample Size Increase

Compare confidence intervals from Steps 6 and 7: - Increasing the sample size makes the results from both methods nearly identical. - The conservativeness of the t-distribution is less pronounced, making it align closely with the normal distribution as sample size increases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Student's t-distribution

The Student's t-distribution is essential when we want to compute confidence intervals and our population standard deviation, \( \sigma \), is unknown. This distribution is particularly useful when working with small sample sizes or when \( \sigma \) isn't directly available. Here's why it's relevant:

• Degrees of Freedom: The concept of degrees of freedom comes into play when using the t-distribution. It's calculated as the sample size minus one (\( n-1 \)), which affects the shape of the distribution.
• Heavier Tails: The t-distribution has heavier tails compared to the standard normal distribution, meaning it accounts for more variability and uncertainty, especially with smaller samples.
• Conservative Intervals: Because of its variability, confidence intervals derived from the t-distribution tend to be more conservative (i.e., wider), effectively covering the parameter more reliably when sample sizes are small.

The key takeaway is that when sample sizes are small and \( \sigma \) is unknown, the Student's t-distribution provides a reliable method for determining confidence intervals by factoring in a bit more uncertainty.

Standard Normal Distribution

The standard normal distribution, often represented by the familiar "bell curve," is a fundamental building block in statistics. It's used when the population standard deviation is known or when our sample size is large enough that the sample standard deviation is a reliable estimate. Key points include:

• Z-scores: This distribution is standardized, with a mean of 0 and a standard deviation of 1. Z-scores are used to determine how far away a data point is from the mean in terms of standard deviations.
• Symmetry and Simplicity: The standard normal distribution is symmetrical and centered around its mean, making it simpler to calculate probabilities and critical values for confidence intervals.
• Connection for Large Samples: For large sample sizes (usually \( n \geq 30 \)), the central limit theorem suggests that the sample mean distribution approximates normality, allowing us to use this distribution confidently.

Thus, when conditions are right, the standard normal distribution offers an efficient method for calculating confidence intervals due to its straightforward and symmetric nature.

Sample Size

Sample size plays a crucial role in determining which distribution is appropriate for calculating confidence intervals. Here's how it affects the choice between the t-distribution and the standard normal distribution:

• Small Sample Size: When the sample size is small (usually \( n < 30 \)), the t-distribution is preferred because it accounts for the additional uncertainty due to unknown \( \sigma \).
• Large Sample Size: As the sample size increases, the sample closely follows the properties of the population distribution. When \( n \geq 30 \), using the sample standard deviation \( s \) as an estimate for \( \sigma \) becomes reliable, and the standard normal distribution can be used.
• Increasing Sample Size: As seen with \( n = 81 \) in the given exercise, larger sample sizes help the t-distribution's results converge with those of the standard normal distribution, meaning smaller intervals and less conservativeness.

The decision on which distribution to use is heavily influenced by the size of the sample. Larger sizes typically allow for simpler computations using the standard normal distribution.