Question

the probability distribution of anas score x on a randomly selected roll of
the ball is shown here.

score	10	20	30	40	50

(a) make a histogram of the probability distribution on paper. describe its shape.
the graph is blank with blank at blank.

(b) calculate and interpret the mean of x.
\\(\bigcirc\\) if ana rolls many, many skee balls, her average score per roll will be about 23.8.
\\(\bigcirc\\) if ana rolls many, many skee balls, her average score per roll will be about 30.
\\(\bigcirc\\) if ana rolls one more skee ball, her score will be about 23.8.
\\(\bigcirc\\) if ana rolls one more skee ball, her score will be about 30.

(c) calculate and interpret the standard deviation of x.
\\(\bigcirc\\) if ana rolls many, many skee balls, the score she earns from a randomly selected skee ball roll typically varies
from the mean by about 159.6 points.
\\(\bigcirc\\) if ana rolls many, many skee balls, the score she earns from a randomly selected skee ball roll typically varies
from the mean by about 12.6 points.
\\(\bigcirc\\) if ana rolls many, many skee balls, the score she earns from a randomly selected skee ball roll will typically be
about 12.6 points.
\\(\bigcirc\\) if ana rolls many, many skee balls, the score she earns from a randomly selected skee ball roll will typically be
about 159.6 points.

Explanation:

Part (a)

Step1: Analyze Probability Distribution

The scores are 10, 20, 30, 40, 50 with probabilities 0.32, 0.27, 0.19, 0.15, 0.07. As the score increases, the probability decreases.

Step2: Determine Histogram Shape

A histogram with decreasing probabilities as the variable (score) increases is right - skewed? No, wait, when the left - most bar (lowest score) has the highest probability and then probabilities decrease as score increases, the distribution is right - skewed? Wait, no. Wait, the score 10 has the highest probability (0.32), then 20 (0.27), 30 (0.19), 40 (0.15), 50 (0.07). So the tail is on the right (higher scores) and the peak is at the left (lower score). So the shape is right - skewed? Wait, no, actually, when the frequency (probability) decreases as the value of the variable increases, the distribution is right - skewed? Wait, no, let's think again. The mode (highest probability) is at 10, then probabilities decrease as score increases. So the graph is right - skewed? Wait, no, the direction of skewness: if the tail is on the right (higher values), it's right - skewed. Here, the probabilities for higher scores (40, 50) are lower, so the tail is on the right. So the graph is right - skewed with the highest probability at 10.

Step1: Recall Mean Formula for Discrete Random Variable

The mean $\mu$ of a discrete random variable $X$ is given by $\mu=\sum xP(x)$, where $x$ is the value of the random variable and $P(x)$ is the corresponding probability.

Step2: Calculate the Mean

We have $x = 10,20,30,40,50$ and $P(x)=0.32,0.27,0.19,0.15,0.07$ respectively.

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The mean of a random variable represents the long - run average value of the variable when the experiment (rolling the skee - ball) is repeated many times. So if Ana rolls many, many Skee Balls, her average score per roll will be about 23.8.

Step1: Recall Standard Deviation Formula for Discrete Random Variable

The standard deviation $\sigma$ of a discrete random variable $X$ is given by $\sigma=\sqrt{\sum(x - \mu)^2P(x)}$, where $\mu$ is the mean, $x$ is the value of the random variable, and $P(x)$ is the corresponding probability. We already know that $\mu = 23.8$.

Step2: Calculate $(x-\mu)^2P(x)$ for each $x$

• For $x = 10$: $(10 - 23.8)^2\times0.32=(- 13.8)^2\times0.32 = 190.44\times0.32 = 60.9408$
• For $x = 20$: $(20 - 23.8)^2\times0.27=(-3.8)^2\times0.27 = 14.44\times0.27 = 3.8988$
• For $x = 30$: $(30 - 23.8)^2\times0.19=(6.2)^2\times0.19 = 38.44\times0.19 = 7.3036$
• For $x = 40$: $(40 - 23.8)^2\times0.15=(16.2)^2\times0.15 = 262.44\times0.15 = 39.366$
• For $x = 50$: $(50 - 23.8)^2\times0.07=(26.2)^2\times0.07 = 686.44\times0.07 = 48.0508$

Step3: Calculate the Variance

Sum up these values: $60.9408+3.8988 + 7.3036+39.366+48.0508=159.56$ (approx). Then the variance $\sigma^{2}=159.56$, and the standard deviation $\sigma=\sqrt{159.56}\approx12.6$
The standard deviation of a random variable represents the typical amount by which the values of the random variable deviate from the mean in the long - run (when the experiment is repeated many times). So if Ana rolls many, many Skee balls, the score she earns from a randomly selected Skee ball roll typically varies from the mean by about 12.6 points.

Answer:

(for part a):
The graph is right - skewed with the highest probability at 10.

Part (b)