Cost of Braces The average cost for teeth straightening with metal braces is approximately \( \$ 5496 \). A nationwide franchise thinks that its cost is below that figure. A random sample of 28 patients across the country had an average cost of \( \$ 5157 \) with a standard deviation of \( \$ 628 \). At \( \alpha=0.005 \), can it be concluded that the mean is less than \( \$ 5496 \) ? Assume that the population is approximately normally distributed. Part: \( 0 / 5 \) Part 1 of 5 (a) State the hypotheses and identify the claim. \[ \begin{array}{l} H_{0}=\mu(\text { Choose one }) ~ \$ \square(\text { Choose one) T } \\ H_{1}=\mu(\text { Choose one } T \$ \text { (Choose one) } T \end{array} \] This hypothesis test is a \( \square \) (Choose one) test.

Let's get into the nitty-gritty of hypothesis testing! Firstly, based on the details provided, we can establish our null and alternative hypotheses. The null hypothesis (\( H_0 \)) suggests that the average cost of braces is \( \$5496 \) (or higher), while the alternative hypothesis (\( H_1 \)) claims that the average cost is actually less than \( \$5496 \). So, we can write: \[ \begin{array}{l} H_{0}:\mu \geq \$5496 \\ H_{1}:\mu < \$5496 \end{array} \] This hypothesis test is a **left-tailed** test because we are looking to see if the mean is less than a certain benchmark. Now, let's keep the fun rolling with some important context! Turns out, braces have a history that’s almost as straight as your future smile! Ancient civilizations, such as the Etruscans in Italy, used rudimentary techniques to straightened teeth over 2000 years ago. Fast forward to the 18th century, and science began to embrace orthodontics as a field with notable figures like Pierre Fauchard paving the way for modern brace designs! Also, while the cost of braces can feel overwhelming, understanding the real-world financial implications can help patients navigate orthodontic care. For example, many dental insurance plans offer coverage for braces, so researching available options might significantly reduce out-of-pocket expenses. Additionally, some clinics provide financing plans that allow patients to pay monthly, making it accessible and more manageable!