The following data points represent how many tunnels Gary the Groundhog dug each year.
$$
18,5,13,9,6,2,10
$$

Using the data, create a histogram.

Question

The following data points represent how many tunnels Gary the Groundhog dug each year.
$$
18,5,13,9,6,2,10
$$

Using the data, create a histogram.

Accepted Answer

∻Solution by Steps∻ ### Creating a Histogram ‖ step 1 ⋮ First, we need to determine the range of the data points. The given data points are: $18, 5, 13, 9, 6, 2, 10$. The minimum value is $2$ and the maximum value is $18$. ‖ ‖ step 2 ⋮ Next, we decide on the number of bins for the histogram. A common choice is to use the square root of the number of data points. Here, we have 7 data points, so $ \sqrt{7} \approx 2.65 $. We can round this to 3 bins. ‖ ‖ step 3 ⋮ Calculate the bin width. The range of the data is $18 - 2 = 16$. Dividing this by the number of bins (3), we get a bin width of $ \frac{16}{3} \approx 5.33 $. We can round this to 5 for simplicity. ‖ ‖ step 4 ⋮ Define the bin intervals. Starting from the minimum value (2), the bins will be: - Bin 1: 2 to 7 - Bin 2: 7 to 12 - Bin 3: 12 to 17 - Bin 4: 17 to 22 ‖ ‖ step 5 ⋮ Count the number of data points in each bin: - Bin 1 (2 to 7): $2, 5, 6$ (3 data points) - Bin 2 (7 to 12): $9, 10$ (2 data points) - Bin 3 (12 to 17): $13$ (1 data point) - Bin 4 (17 to 22): $18$ (1 data point) ‖ ‖ step 6 ⋮ Draw the histogram with the bins on the x-axis and the frequency (number of data points) on the y-axis. ‖ ∻Answer∻ ⚹ The histogram will have 4 bins with the following frequencies: Bin 1: 3, Bin 2: 2, Bin 3: 1, Bin 4: 1 ⚹ ### Determining the Smallest Sample Size for a Desired Margin of Error ‖ step 1 ⋮ To determine the smallest sample size $ n $ for a desired margin of error $ E $, we use the formula: $$ n = \left( \frac{Z \cdot \sigma}{E} \right)^2 $$ where $ Z $ is the Z-score corresponding to the desired confidence level, $ \sigma $ is the population standard deviation, and $ E $ is the margin of error. ‖ ‖ step 2 ⋮ Identify the values for $ Z $, $ \sigma $, and $ E $. For example, if we want a 95% confidence level, $ Z \approx 1.96 $. Suppose $ \sigma = 10 $ and $ E = 2 $. ‖ ‖ step 3 ⋮ Substitute the values into the formula: $$ n = \left( \frac{1.96 \cdot 10}{2} \right)^2 = \left( \frac{19.6}{2} \right)^2 = 9.8^2 = 96.04 $$ Since the sample size must be a whole number, we round up to 97. ‖ ∻Answer∻ ⚹ The smallest sample size required is 97 ⚹ ### Calculating the Mode of a Set of Numbers ‖ step 1 ⋮ The mode is the number that appears most frequently in a data set. Given the data points: $18, 5, 13, 9, 6, 2, 10$, we count the frequency of each number. ‖ ‖ step 2 ⋮ Since all numbers appear only once, there is no mode in this data set. ‖ ∻Answer∻ ⚹ There is no mode in the given data set ⚹ ### Finding the Mean and Standard Deviation of the Distribution ‖ step 1 ⋮ To find the mean, sum all the data points and divide by the number of data points: $$ \text{Mean} = \frac{18 + 5 + 13 + 9 + 6 + 2 + 10}{7} = \frac{63}{7} = 9 $$ ‖ ‖ step 2 ⋮ To find the standard deviation, first calculate the variance. The variance is the average of the squared differences from the mean: $$ \text{Variance} = \frac{(18-9)^2 + (5-9)^2 + (13-9)^2 + (9-9)^2 + (6-9)^2 + (2-9)^2 + (10-9)^2}{7} $$ $$ = \frac{81 + 16 + 16 + 0 + 9 + 49 + 1}{7} = \frac{172}{7} \approx 24.57 $$ ‖ ‖ step 3 ⋮ The standard deviation is the square root of the variance: $$ \text{Standard Deviation} = \sqrt{24.57} \approx 4.96 $$ ‖ ∻Answer∻ ⚹ The mean is 9 and the standard deviation is approximately 4.96 ⚹ ### Key Concepts and Explanations ∻Key Concept∻ ⚹Histogram creation involves determining bin intervals and counting data points within those intervals⚹ ∻Explanation∻ ⚹A histogram visually represents the frequency distribution of a data set by grouping data points into bins⚹ ∻Key Concept∻ ⚹Sample size determination uses the margin of error formula involving Z-score and standard deviation⚹ ∻Explanation∻ ⚹The formula helps in calculating the minimum sample size needed to achieve a desired margin of error at a specific confidence level⚹ ∻Key Concept∻ ⚹The mode is the most frequently occurring number in a data set⚹ ∻Explanation∻ ⚹If no number repeats, the data set has no mode⚹ ∻Key Concept∻ ⚹Mean and standard deviation are measures of central tendency and dispersion⚹ ∻Explanation∻ ⚹The mean is the average of the data points, while the standard deviation measures the spread of the data points around the mean⚹◊What is the frequency of the number of tunnels dug by Gary the Groundhog each year?⍭ Generate me a similar question◊

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