You research the salaries of senior-level civil engineers and find that the population mean is \$131,935. In Exercise 4, does the t-value fall between -t0.95 and t0.95?

Verified step by step guidance

1. Identify the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis typically states that there is no significant difference, while the alternative hypothesis suggests there is a difference.
2. Determine the sample mean, sample size, and sample standard deviation from Exercise 4 (these values are necessary to calculate the t-value).
3. Calculate the t-value using the formula: $t=\frac{\left(x_{-}\mu \right)}{\frac{s_{}}{\sqrt{n}}}$, where x is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.
4. Find the critical t-values for a 95% confidence level (t0.95 and -t0.95) using a t-distribution table or statistical software. The degrees of freedom (df) are calculated as $\mathrm{df}=n-1$.
5. Compare the calculated t-value to the critical t-values. If the t-value falls between -t0.95 and t0.95, the null hypothesis is not rejected; otherwise, it is rejected.

Key Concepts

Population Mean

The population mean is the average of a set of values in a complete population. In this context, it represents the average salary of all senior-level civil engineers, which is $131,935. Understanding the population mean is crucial for making inferences about the data and comparing it to sample statistics.

t-Distribution

The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. It is used in statistics when the sample size is small or the population standard deviation is unknown. The t-values, such as -t0.95 and t0.95, represent critical values that help determine the range of values for hypothesis testing.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis and an alternative hypothesis, then using test statistics (like t-values) to determine whether to reject the null hypothesis. In this case, checking if the t-value falls between -t0.95 and t0.95 helps assess the significance of the sample mean in relation to the population mean.