Figure 5-6 shows histograms of several binomial...

Figure $5-6$ shows histograms of several binomial distributions with $n=6$ trials. Match the given probability of success with the best graph. (a) $p=0.30$ goes with graph (b) $p=0.50$ goes with graph (c) $p=0.65$ goes with graph (d) $p=0.90$ goes with graph (e) In general, when the probability of success $p$ is close to $0.5,$ would you say that the graph is more symmetric or more skewed? In general, when the probability of success $p$ is close to $1,$ would you say that the graph is skewed to the right or to the left? What about when $p$ is close to $0 ?$

Figure $5-6$ shows histograms of several binomial distributions with $n=6$ trials. Match the given probability of success with the best graph.
(a) $p=0.30$ goes with graph
(b) $p=0.50$ goes with graph
(c) $p=0.65$ goes with graph
(d) $p=0.90$ goes with graph
(e) In general, when the probability of success $p$ is close to $0.5,$ would you say that the graph is more symmetric or more skewed? In general, when the probability of success $p$ is close to $1,$ would you say that the graph is skewed to the right or to the left? What about when $p$ is close to $0 ?$

Key Concepts

Binomial Distribution

A binomial distribution describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) and the probability of success remains constant across trials. It is characterized by two parameters: n, the number of trials, and p, the probability of success in each trial.

Probability (p) and Its Influence on Shape

The value of p in a binomial distribution critically affects the shape of its histogram. When p is low, most outcomes tend to be failures and the distribution is skewed. In contrast, when p is high, almost all outcomes are successes and similar skewness occurs in the opposite direction. When p is near 0.5, outcomes are more evenly distributed, resulting in a more symmetric histogram.

Symmetry and Skewness in Distributions

Symmetry in a binomial distribution is most evident when p is around 0.5, as successes and failures are equally likely, leading to a balanced distribution around the mean. Skewness occurs when p deviates from 0.5; specifically, the histogram is skewed to the right when p is small (long tail on the right) and skewed to the left when p is large (long tail on the left). These concepts help in predicting and understanding the visual distribution of outcomes.

Transcript

00:01 In question number 8, in point a, given that n equal to 6, and p equal to 0 .30, the expected value of binomial distribution is the product of the number of trails in under probability p, so it should be equal to n by p equal to 6 by 0 .30 equal to 1 .80.

00:25 The highest bars of the histogram should be about the expect value.

00:36 Thus, the second and the third bar should be the highest, which corresponds with the graph 2.

01:00 Note that the first bar of the histogram corresponds with r equal to 0, and the second corresponds with r equal to 1 and the third bar corresponds with r equal to 2 and at c in point b if it was given that in equal to 6 and p equal to 0 .50 so the expected value will be np equal to 6 by 0 .50 equal to 3.

01:58 So the highest bars of the histogram should be about the expect value.

02:04 Thus the fourth bar, which corresponds with r equal to 3, should be the highest, which corresponds with graph 1.

02:26 Note that the first bar of the histograms, the instagram corresponds with r equal to 0 and the second corresponds with r equal to 1 and the third corresponds with r equal to an atc.

02:40 In point c, if it's given that n equal to 6 and p equal to 0 .65, so the expected value of a binomial distribution is the product of the number of trails and under probability b.

03:01 So it will be equal to np equal to 6 by 0 .65 approximately equal to 4.

03:12 The highest bars of the histogram should be about the expected value, thus the fifth bar, which corresponds with r equal to z, equal to 4, should be the highest, which corresponds with graph triple i.

03:40 That the first bar of the histogram corresponds with r equal to 0, the second corresponds with r equal to 1 and the third with r equal to 2...

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