You research prices of cell phones and find that the population mean is \$431.61. In Exercise 19, does the t-value fall between -t0.95 and t0.95?

Verified step by step guidance

1. Step 1: Understand the problem. The question is asking whether the calculated t-value falls within the range defined by the critical t-values (-t₀.₉₅ and t₀.₉₅) for a given confidence level (95%). This involves hypothesis testing and comparing the t-value to the critical values.
2. Step 2: Identify the formula for the t-value. The t-value is calculated using the formula: $t=\frac{x-\mu }{\frac{s}{\sqrt{n}}}$, where $x$ is the sample mean, $\mu$ is the population mean, $s$ is the sample standard deviation, and $n$ is the sample size.
3. Step 3: Determine the critical t-values (-t₀.₉₅ and t₀.₉₅). These values depend on the degrees of freedom (df), which is calculated as $\mathrm{df}=n-1$. Use a t-distribution table or statistical software to find the critical values for a 95% confidence level.
4. Step 4: Calculate the t-value using the formula from Step 2. Plug in the sample mean, population mean, sample standard deviation, and sample size into the formula to compute the t-value.
5. Step 5: Compare the calculated t-value to the critical values (-t₀.₉₅ and t₀.₉₅). If the t-value falls within this range, it is within the 95% confidence interval. Otherwise, it is outside the interval.

Key Concepts

Population Mean

The population mean is the average value of a set of data points in a complete population. It is calculated by summing all the values and dividing by the number of values. In this context, the population mean of $431.61 represents the average price of cell phones across the entire population being studied.

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 define the boundaries for hypothesis testing.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters 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. The comparison of the calculated t-value to critical values helps assess the significance of the results.