Question

A data scientist tracks the number of songs that are downloaded from a particular website over the course of a year. She used the data to build a probability distribution where the random variable X represents the number of songs downloaded per person per day. The distribution is given below. Find the mean and the standard deviation of the probability distribution using a TI-83, TI-83 Plus, or TI-84 graphing calculator. Round the mean to two decimal places and the standard deviation to three decimal places. x P(x) 0 0.30 1 0.02 2 0.03 3 0.04 4 0.03 5 0.04 6 0.02 7 0.01 8 0.01 9 0.25 10 0.25

Answer 1

The mean of the probability distribution is μ = 4.17 (rounded to two decimal places), and the standard deviation is σ = 2.432 (rounded to three decimal places).

Step-by-step explanation:

To find the mean
`(\( \mu \))` , we multiply each value of the random variable x by its corresponding probability
`(\( P(x) \))` , sum these products, and round the result to two decimal places. For this distribution:

`\[ \mu = 0(0.30) + 1(0.02) + 2(0.03) + 3(0.04) + 4(0.03) + 5(0.04) + 6(0.02) + 7(0.01) + 8(0.01) + 9(0.25) + 10(0.25) \]`

Calculating this expression yields
`\( \mu = 4.17 \`).

To find the standard deviation
`(\( \sigma \)),` we use the formula:

`\[ \sigma = \sqrt{\sum_(i=1)^(n) P(x_i) \cdot (x_i - \mu)^2} \]`

Here,
`\( x_i \)` represents each value of the random variable,
`( \mu \)` is the mean, and
`\( P(x_i) \)` is the corresponding probability. After performing the calculations, rounding the result to three decimal places, we find
`\sigma = 2.432 \).`

In summary, the mean and standard deviation provide important insights into the central tendency and variability of the data. The mean indicates the average number of songs downloaded per person per day, while the standard deviation quantifies the dispersion around this mean. These statistics are valuable for understanding the characteristics of the probability distribution and making informed interpretations based on the data collected by the data scientist.

Answer 2

The standard deviation of the probability distribution is
`\( 4.288 \)` when rounded to three decimal places.

To calculate the mean and standard deviation of the given probability distribution using a TI-83, TI-83 Plus, or TI-84 grap_hing calculator, follow these steps:

1. Turn on your calculator and press the `STAT` button.

2. Select `EDIT` by pressing `1` or by simply pressing `ENTER` if `1:Edit...` is already highlighted.

3. Enter the values of
`\( x \)` into the `L1` list. These are the number of songs downloaded per person per day.

4. Enter the corresponding probabilities
`\( P(x) \)` into the `L2` list.

5. Once the data is entered, press the `STAT` button again and move right to the `CALC` menu.

6. Select `1-Var Stats` by pressing `1` or by simply pressing `ENTER` if `1:1-Var Stats` is already highlighted.

7. If the data for
`\( x \)` is in `L1` and the data for
`\( P(x) \)` is in `L2`, then type `L1, L2` (you can access the `L` variables by pressing the `2ND` button and then the corresponding number).

8. Press `ENTER` to calculate the statistics.

9. The calculator will display several statistics, including the mean and the standard deviation (denoted by `σx` for a population or `S_x` for a sample).

For the mean, sum up the products of each value of
`\( x \)` and its corresponding probability
`\( P(x) \)`:

`\[ \text{Mean} (\mu) = \sum (x \cdot P(x)) \]`

For the standard deviation, use the formula:

`\[ \text{Standard Deviation} (\sigma) = √(\sum \left[(x - \mu)^2 \cdot P(x)\right]) \]`

The mean of the probability distribution is
`\( 5.54 \)` when rounded to two decimal places.

The standard deviation of the probability distribution is
`\( 4.288 \)` when rounded to three decimal places.