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When dealing with confidence intervals, particularly with small sample sizes (typically less than 30), the t-distribution is the go-to method. Unlike the normal distribution, which uses the same standard curve regardless of sample size, the t-distribution adjusts its shape depending on the size of the sample. This flexibility makes it more accurate for small sample sizes.
The t-distribution is similar to the normal distribution; however, it has thicker tails. This means that it tends to spread out more, allowing for more variability and helping to account for the greater uncertainty found in smaller samples.

• The thicker tails mean that the standard deviation of the t-distribution is wider.
• This is why we use a t-distribution when finding confidence intervals for small sample sizes.
• As the sample size increases, the t-distribution approaches the normal distribution.

In statistics, the sample mean is a crucial element when estimating population parameters. It represents the average value of a sample taken from a larger population. The sample mean is denoted as \( \bar{x} \), and in this exercise, it was given as 24.

• The sample mean serves as an unbiased estimator of the population mean due to the Central Limit Theorem.
• It provides a point estimate for the average number of times houses were shown in our exercise scenario.

By using the sample mean, we are making an assumption that it reflects the central tendency of the entire population, even though we only examined a small part of it.

The margin of error (ME) helps us understand the potential error in our estimate of a population parameter. It is an indication of how much we can expect the sample estimate to differ from the true population value. In this context, it's calculated using the t-distribution critical value and the sample's standard deviation. To calculate the margin of error, use:\[ ME = t^* \times \frac{s}{\sqrt{n}} \]Where:\( t^* \) is the critical value from the t-distribution (in this case, approximately 2.977),
\( s \) is the sample standard deviation (here, 5),
\( n \) is the sample size (15 houses in this example).The margin of error results in a numeric range around the sample mean, which is added and subtracted to form the confidence interval.

Degrees of freedom (df) play a critical role in statistical calculations. They represent the number of values that have the freedom to vary within a statistical calculation. For confidence intervals using t-distribution, the degrees of freedom are calculated as \( n - 1 \).

• In our particular exercise, with a sample size of 15, the degrees of freedom are \( 15 - 1 = 14 \).
• Degrees of freedom affect the critical value \( t^* \) used in forming confidence intervals.
• A higher number of degrees of freedom results in a t-distribution that more closely resembles a normal distribution.

Understanding degrees of freedom is essential since they help determine the shape of the t-distribution, which aids in calculating an accurate confidence interval.