For a confidence level of $90 \%$ with a sample size of 40, find the critical value $t_c$.
Round to 4 decimal places:
$t_c=$
To determine the critical value \(t_c\) for a \(90\%\) confidence level, we must consider the characteristics of the t-distribution. Since the population standard deviation is unknown, we use the t-distribution with degrees of freedom calculated as \(df = n - 1\). Given a sample size \(n = 40\), the degrees of freedom are \(df = 40 - 1 = 39\). A \(90\%\) confidence level means that the total area in the two tails of the distribution is \(\alpha = 1 - 0.90 = 0.10\). Because we are looking for the critical value associated with this confidence interval, we divide this significance level equally between the two tails, yielding \(\alpha / 2 = 0.05\) for each tail. Our task is to find the value \(t_c\) such that the area in the upper tail is \(0.05\), which is equivalent to finding the value at the \(0.95\) cumulative probability point for a t-distribution with \(39\) degrees of freedom.
Using the t-distribution properties, we find the critical value \(t_c\) for \(df = 39\) at a cumulative probability of \(0.95\). Calculating this quantile, we obtain a value approximately equal to \(1.684875\). Rounding this result to four decimal places, we get \(t_c \approx 1.6849\). This value represents the boundary that marks the top \(5\%\) of the distribution, which is essential for constructing the confidence interval for the mean.
When researchers perform a study, such as measuring the average height of a group of students, they often don't know the population's true standard deviation. If they have a sample size of \(n=40\), they use the t-distribution to account for the uncertainty. To construct a \(90\%\) confidence interval for the population mean \(\mu\), they calculate the critical value \(t_c\), which is then used in the formula \(\bar{x} \pm t_c \times \frac{s}{\sqrt{n}}\). This ensures that the range calculated captures the true mean with \(90\%\) confidence.