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Confidence Interval for \(\mu_{1}-\mu_{2}\) Consider two independent normal distributions. A random sample of size \(n_{1}=20\) from the first distribution showed \(\bar{x}_{1}=12\) and a random sample of size \(n_{2}=25\) from the second distribution showed \(\bar{x}_{2}=14\) (a)If \(\sigma_{1}\) and \(\sigma_{2}\) are known, what distribution does \(\bar{x}_{1}-\bar{x}_{2}\) follow? Explain. (b) Given \(\sigma_{1}=3\) and \(\sigma_{2}=4,\) find a \(90 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) (c) Suppose \(\sigma_{1}\) and \(\sigma_{2}\) are both unknown, but from the random samples, you know \(s_{1}=3\) and \(s_{2}=4 .\) What distribution approximates the \(\bar{x}_{1}-\bar{x}_{2}\) distribution? What are the degrees of freedom? Explain. (d) With \(s_{1}=3\) and \(s_{2}=4,\) find a \(90 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) (e) If you have an appropriate calculator or computer software, find a \(90 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) using degrees of freedom based on Satterthwaite's approximation. (f) Based on the confidence intervals you computed, can you be \(90 \%\) confident that \(\mu_{1}\) is smaller than \(\mu_{2} ?\) Explain.

Step by step solution

01

Understand the Distribution for Known Variances

Since both populations are normally distributed and the population variances \( \sigma_1 \) and \( \sigma_2 \) are known, the difference \( \bar{x}_1 - \bar{x}_2 \) will follow a normal distribution. This is due to the Central Limit Theorem, which holds even for small sample sizes when populations are normally distributed.

02

Calculate the 90% Confidence Interval with Known Variances

The standard error \(SE\) for the difference in means is calculated as \[ SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} = \sqrt{\frac{3^2}{20} + \frac{4^2}{25}} = \sqrt{0.45 + 0.64} = \sqrt{1.09} \approx 1.044. \] The critical value for a 90% confidence interval is 1.645 (Z-value). Thus, the confidence interval is \( (\bar{x}_1 - \bar{x}_2) \pm 1.645 \times SE \). Calculating gives \[ (12 - 14) \pm 1.645 \times 1.044 = -2 \pm 1.717. \] So the interval is approximately \((-3.717, -0.283)\).

03

Distribution for Unknown Variances

When both \( \sigma_1 \) and \( \sigma_2 \) are unknown, the distribution of \( \bar{x}_1 - \bar{x}_2 \) can be approximated by a t-distribution. The approximate degrees of freedom (\(df\)) can be calculated using Satterthwaite's approximation.

04

Calculate Approximate Degrees of Freedom

Satterthwaite's approximation for \(df\) is \[ df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{(s_1^2/n_1)^2}{n_1 - 1} + \frac{(s_2^2/n_2)^2}{n_2 - 1}} \] which equals \[ df = \frac{(\frac{9}{20} + \frac{16}{25})^2}{\frac{(9/20)^2}{19} + \frac{(16/25)^2}{24}} \approx 36.2. \] You would typically round down to the nearest whole number, giving 36 degrees of freedom.

05

Compute the Confidence Interval for Unknown Variances

Using the t-distribution with 36 degrees of freedom, we find a critical value of approximately 1.690 for a 90% confidence level. The confidence interval is calculated as \( (\bar{x}_1 - \bar{x}_2) \pm 1.690 \times SE \), where \( SE = 1.044 \). Thus \[ -2 \pm 1.690 \times 1.044 = -2 \pm 1.763. \] The interval is approximately \((-3.763, -0.237)\).

06

Confidence Interval using Satterthwaite's Approximation

Using computer software to apply Satterthwaite's approximation, the 90% confidence interval for \( \mu_1 - \mu_2 \) might show slight variations around the hand-calculated interval due to nuances in the computational approach.

07

Interpret the Confidence Intervals

All calculated confidence intervals for \( \mu_1 - \mu_2 \) do not include zero and are entirely negative, suggesting with 90% confidence that \( \mu_1 \) is less than \( \mu_2 \).

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Key Concepts

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

Normal Distribution

In statistics, a normal distribution is a common probability distribution characterized by its bell-shaped curve, often called a Gaussian distribution. When you have large samples or distributions established to be normal, such as in the case where the variances are known and populations are normal, the difference in sample means, noted as \( \bar{x}_1 - \bar{x}_2 \), will itself follow a normal distribution.

This allows us to utilize the normal distribution directly to create confidence intervals if we know the standard deviations of the two populations. You can calculate the confidence interval using the z-score, which represents standard deviations away from the mean in a normal distribution.

For example, with a 90% confidence level, you may use a z-value of 1.645 to determine the range in which the true difference in population means is expected to fall.

Normal distribution assumptions become crucial because they simplify calculations and provide a strong foundation for understanding the probability of certain outcomes based upon sample data. Understanding this foundation lets us confidently make predictions and interpretations.

T-Distribution

A t-distribution is closely related to the normal distribution but is used in scenarios where sample sizes are small, and population variances are unknown. This distribution is wider and fatter-tailed, reflecting the greater uncertainty when estimates are based on small samples.

When determining a confidence interval for \( \bar{x}_1 - \bar{x}_2 \) with unknown variances, we approximate the distribution with a t-distribution. This involves using sample standard deviations \( s_1 \) and \( s_2 \) instead of the known population standard deviations \( \sigma_1 \) and \( \sigma_2 \).

For calculations involving two samples, you'll use a specific critical value from the t-distribution, determined by degrees of freedom which influences the distribution's shape. When sample sizes are small or variances are not known, the t-distribution provides a more accurate reflection of the data's potential variability than narrowly estimating through a normal distribution.

Grasping t-distribution concepts is critical when working with real-world data, where assumptions about population parameters often cannot be precisely known.

Satterthwaite's Approximation

When dealing with t-distributions for the difference between two means, calculating the degrees of freedom is slightly more complex due to the variability introduced by having two sample standard deviations. Satterthwaite's approximation provides a method to determine these approximate degrees of freedom.

This approach combines the variances of the two samples, adjusted by their respective sizes, to result in a single number representing the overall degrees of freedom for the comparison. The formula might look complex but effectively unifies the uncertainty propagated from estimating two separate population measures with limited data.

Using Satterthwaite's approximation permits more refined use of the t-distribution in constructing confidence intervals. By reflecting the actual variability more accurately than simpler methods, it ensures the confidence intervals calculated are as precise as possible given the available information.

It’s essential to use tools or software that can handle these calculations when your datasets become complex, ensuring your statistical conclusions are based on sound mathematical footing.

Central Limit Theorem

The Central Limit Theorem (CLT) is a fundamental theorem in statistics. It states that, given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples will be approximately normally distributed. This holds true regardless of the actual distribution of the population being studied.

In our case, when sample means \( \bar{x}_1 \) and \( \bar{x}_2 \) come from independent normal populations, CLT justifies using normal approximation for smaller sample sizes because the population was initially normal. This theorem enables statisticians to use what they know about normal distributions to make inferences about sample means even when dealing with variations in sample size.

As sample sizes grow, the approximation becomes even more accurate, making CLT both powerful and versatile, a cornerstone of practical applied statistics. Understanding CLT aids in explaining why certain central tendencies can be assumed even when full population data is unavailable, making statistical analyses of real-world data more feasible.