Question

The following questions are from ARTIST (reproduced with permission) One of the items on the student survey for an introductory statistics course was "Rate your intelligence on a scale of 1 to \(10 . "\) The distribution of this variable for the 100 women in the class is presented below. What is the probability of randomly selecting a woman from the class who has an intelligence rating that is LESS than seven (7)? $$ \begin{array}{|c|c|} \hline \text { Intelligence Rating } & \text { Count } \\ \hline 5 & 12 \\ \hline 6 & 24 \\ \hline 7 & 38 \\ \hline 8 & 23 \\ \hline 9 & 2 \\ \hline 10 & 1 \\ \hline \end{array} $$ a. \((12+24) / 100=.36\) b. \((12+24+38) / 100=.74\) c. \(38 / 100=.38\) d. \((23+2+1) / 100=.26\) e. None of the above.

Short answer

a. (12+24)/100=0.36

Step-by-step solution

Identify Relevant Information

We are given a table of intelligence ratings and their corresponding counts for women in a statistics class. We want to find the probability of selecting a woman with an intelligence rating less than 7.

Determine Target Counts

From the table, the ratings less than 7 are 5 and 6. The counts for these ratings are 12 for a rating of 5 and 24 for a rating of 6.

Calculate Total of Target Counts

Add the counts for the ratings less than 7: \(12 + 24 = 36\).

Calculate Probability

Divide the total of the target counts by the total number of women: \( \frac{36}{100} = 0.36 \).

Choose the Correct Answer

The probability calculated is 0.36, which corresponds to option a: \((12+24)/100=0.36\).

Key concepts

Descriptive Statistics

Descriptive statistics refers to tools and techniques that provide a summary or insight into the features of a data set. It includes measures like mean, median, mode, range, and others. These measures help us understand the distribution, central tendency, and variability of data.

- **Central Tendency Measures** include: - **Mean:** the average of all data points. - **Median:** the middle point of a data set. - **Mode:** the most frequently occurring data point.
- **Variability Measures** include: - **Range:** the difference between the highest and lowest values. - **Standard Deviation:** a measure of how spread out the data is from the mean.

In the context of the exercise, the intelligence ratings table represents a **frequency distribution** for a sample of 100 women. Descriptive statistics can provide a clearer understanding of how intelligence ratings are spread across these women. For instance, **mode** is a useful concept here since we can observe the most common rating by looking at the frequency counts.

Distributions

Distributions in statistics describe how values in a data set are spread across possible outcomes. This exercise provides an illustration of a simple distribution, namely how intelligence ratings are distributed among 100 women. A distribution can be graphed as a histogram or a plot to visualize how data points are grouped and vary.

Key distribution types include: - **Normal Distribution:** a bell-shaped curve. - **Binomial Distribution:** outcomes of binary (success/failure) experiments. - **Uniform Distribution:** equal probability for all outcomes.

In the given exercise, the distribution of intelligence ratings is categorical. Different ratings have different frequencies, indicating how many women rate themselves at each level from 5 to 10. By examining such distributions, you can evaluate the central trends and outliers, and in this instance, it helps identify the proportion of women with ratings less than 7. The calculation of probability in the exercise involves determining how common a rating below 7 is, among all the data points in this distribution.

Introductory Statistics

Introductory statistics encompass basic principles that lay the foundation for understanding more complex statistical concepts. It often involves learning about types of data, data collection methods, and basic statistical calculations.

Some fundamental topics in introductory statistics include: - **Probability:** assessing the likelihood of an event occurring. - **Data Collection:** understanding sampling methods and survey techniques. - **Basic Graphical Techniques:** such as histograms and boxplots.

In the problem at hand, a crucial aspect is understanding how to calculate probability. This involves practical counting, as demonstrated by figuring out the number of women with intelligence ratings below a specific threshold. By understanding these essential principles, students are better equipped to handle statistical data and conduct basic analyses. Introductory statistics is essential in helping students interpret graphs and data across different contexts, assisting them in making informed decisions based on statistical data.