Constructing Confidence Intervals for μ1-μ2. You can construct a confidence interval for the difference between two population means μ1-μ2 , as shown below, when both population standard deviations are known, and either both populations are normally distributed or both n1>= 30 and n2>=30 . Also, the samples must be randomly selected and independent.
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In Exercises 29 and 30, construct the indicated confidence interval for μ1-μ2 .


Software Engineer Salaries Construct a 95% confidence interval for the difference between the mean annual salaries of entry level software engineers in Santa Clara, California, and Greenwich, CT, using the data from Exercise 27.

Verified step by step guidance

1. Step 1: Identify the given data from the problem. You need the sample means (x̄₁ and x̄₂), population standard deviations (σ₁ and σ₂), sample sizes (n₁ and n₂), and the confidence level (95%).
2. Step 2: Determine the critical value (z*) for a 95% confidence level. Since the population standard deviations are known, use the standard normal distribution (Z-distribution) to find the z* value corresponding to a 95% confidence level.
3. Step 3: Calculate the standard error (SE) for the difference between the two means using the formula: $\sqrt{\frac{{\sigma }^1}{2}+\frac{{\sigma }^2}{2}}$, where σ₁ and σ₂ are the population standard deviations, and n₁ and n₂ are the sample sizes.
4. Step 4: Compute the margin of error (ME) using the formula: $z*\mathrm{SE}$, where z* is the critical value and SE is the standard error calculated in Step 3.
5. Step 5: Construct the confidence interval for μ₁ - μ₂ using the formula: $(\mathrm{x̄}1-\mathrm{x̄}2)\pm \mathrm{ME}$, where x̄₁ and x̄₂ are the sample means, and ME is the margin of error. This will give you the lower and upper bounds of the confidence interval.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is expressed with a certain level of confidence, such as 95%, indicating the probability that the interval includes the parameter. For example, if a 95% confidence interval for the difference between two means is calculated, it suggests that if the same sampling process were repeated multiple times, approximately 95% of the intervals would contain the true difference.

Independent Samples

Independent samples refer to two or more groups of data that are collected separately and do not influence each other. In the context of constructing confidence intervals for the difference between two means, it is crucial that the samples are independent to ensure that the results are valid. For instance, comparing salaries of software engineers from two different cities assumes that the selection of engineers in one city does not affect the selection in the other.

Normal Distribution and Sample Size

The assumption of normal distribution is important when constructing confidence intervals, especially when sample sizes are small. If the populations are normally distributed, the sample means will also be normally distributed. However, if the sample sizes are large (n1 ≥ 30 and n2 ≥ 30), the Central Limit Theorem states that the sampling distribution of the mean will approximate normality regardless of the population distribution, allowing for valid confidence interval construction.