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The Student's t-distribution is particularly useful when we deal with small sample sizes (typically less than 30) or when the population standard deviation is unknown. This distribution is bell-shaped and similar to the normal distribution but has thicker tails to account for the additional uncertainty in smaller samples.
The thicker tails mean there's a higher probability of values further away from the mean, which helps mitigate the risk of misestimating the mean of a small sample.

• Best used when sample size is small and population standard deviation is unknown.
• Has thicker tails than the normal distribution.
• This makes it more adaptable to samples where extreme values might occur.

When having a larger sample size, such as 81 in our exercise, the Student's t-distribution becomes very similar to the standard normal distribution, making it less critical to differentiate between them.

The z-distribution, also known as the standard normal distribution, is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It is typically used in statistics when you have a large sample size, generally over 30.
In our exercise, the sample size is 81, which is considered large enough to use the z-distribution. This is because the larger the sample, the more the distribution of the sample mean will approximate a normal distribution due to the Central Limit Theorem.

• Used primarily for large samples.
• Mean is 0 and standard deviation is 1.
• Assumes the population has a normal distribution or the sample size is sufficiently large.

Since the sample size in our exercise exceeded 30, both the Student's t-distribution and the z-distribution were options, but the z-distribution was chosen for simplicity and because the exact population distribution was unknown.

The critical value is a point on the test distribution that is compared to the test statistic to determine whether to reject the null hypothesis.
In the context of confidence intervals, the critical value defines the margin of error, dictating how wide the interval must be to encapsulate the true population parameter with a given level of confidence.

• For a 95% confidence interval using the z-distribution, the critical value is 1.96.
• The critical value is derived from z-tables or t-tables based on the confidence level.
• Determines the span of the interval above and below the sample mean.

This means for our exercise, the critical value of 1.96 is used to calculate the 95% confidence interval for the population mean, ensuring that the mean lies within a certain range 95% of the time.

Standard Error (SE) is a measure of the variability or dispersion of the sample mean estimate of a population parameter. It helps assess how accurately the sample mean approximates the true population mean.
The standard error decreases as the sample size increases, indicating more precision in the sample's representation of the population.

• Calculated by dividing the sample standard deviation by the square root of the sample size: \[SE = \frac{s}{\sqrt{n}}\]
• Smaller SE means the sample mean is a more accurate estimation of the population mean.
• In our exercise, with a sample size of 81 and standard deviation 3, SE is 0.333, indicating a reasonable degree of precision.

A smaller standard error, as calculated in our exercise, means our sample mean (20) is a reliable estimate of the population mean.