Statistics Quiz with Solutions (5 Questions)

To find the sample mean, which is the average height in this case, you add up all the heights of the students in the sample and then divide that total by the number of students. Here's the simple formula:

\[ \text{Sample Mean} = \frac{\text{Sum of all heights}}{\text{Number of students}} \]

For our example, we added together the heights of the 5 students and divided by 5. This gave us the average height, or sample mean, of 65.3 inches.

A parameter is a numerical value that describes a specific characteristic of a population. In statistics, a population refers to the complete set of items that interest us, which could be people, objects, events, or measurements. Parameters are used to summarize this large set of data into a single value or a set of values that provide insights about the entire population.

Some common examples of parameters include:

- **Mean (Average):** The sum of all values divided by the number of values in the population.
- **Median:** The middle value when the data is arranged in order.
- **Mode:** The value that appears most frequently in the data set.
- **Standard Deviation:** A measure of the amount of variation or dispersion in the set of data values.

Unlike statistics, which are similar measures but derived from a sample of the population, parameters are considered fixed values since they describe the entire population. However, in practice, because it's often not feasible to study every member of a population, parameters are usually estimated through the use of samples and statistical methods.

The goodness-of-fit chi-square (χ²) test is used to determine how well observed data fit an expected distribution. Larger values of χ² indicate a greater deviation from the expected distribution, suggesting that the observed data do not fit well. Therefore, the test is right-tailed because we are typically interested in whether the observed deviation is significantly greater than what would be expected by chance, indicating a poor fit.

The indicated value \(z_{0.02}\) refers to the z-score in a standard normal distribution where there is 0.02 (or 2%) of the area under the curve to the right of this z-score. This value is typically found using a standard normal (Z) table or statistical software, as it corresponds to the 98th percentile.

The indicated value of the standard normal variable \(Z\), \(z_{0.02}\), is approximately 2.05. This means that 2% of the area under the standard normal distribution curve lies to the right of \(z = 2.05\).

The normal distribution is described by the bell curve due to its characteristic shape, which resembles the outline of a bell. This shape arises from the mathematical properties of the normal distribution formula. Here's why the bell curve is a fitting description:

1. **Symmetry:** The normal distribution is perfectly symmetrical around its mean. This means that the left half of the distribution is a mirror image of the right half.

2. **Peak at the Mean:** The highest point of the curve occurs at the mean (\( \mu \)), which is also the median and mode in a normal distribution. This central peak tapers off symmetrically on both sides, decreasing as one moves away from the mean.

3. **Tails:** The curve approaches, but never touches, the horizontal axis. It extends infinitely in both directions, indicating that all values are possible for a normally distributed variable, but values far from the mean are less likely. These "tails" of the distribution decrease asymptotically towards the axis.

4. **Mathematical Equation:** The shape comes from the equation of the normal distribution. The equation involves \( e \) (Euler's number, approximately 2.71828) raised to the power of a negative quadratic function of the variable. This mathematical form ensures the symmetrical, bell-shaped curve.

The bell curve is a fundamental concept in statistics because it describes the expected distribution of many natural phenomena, from heights and test scores to errors in measurements, making it a powerful tool for statistical analysis and probability theory.