Question

the histograms each represent part of a binomial distribution. each distribution has the same probability of success, p, but different numbers of trials, n. identify the unusual values of x in each histogram.
(a) n = 4
(b) n = 8
(c) n = 12

identify the unusual values of x in histogram (b). choose the correct answer below.
a. x = 5
b. x = 0
c. x = 0, x = 1, x = 2, and x = 8
d. there are no unusual values of x in the histogram.

identify the unusual values of x in histogram (c). choose the correct answer below.
a. x = 0, x = 1, x = 2, x = 3, x = 4, x = 11, and x = 12
b. x = 0
c. x = 0, x = 1, x = 2, and x = 8
d. there are no unusual values of x in the histogram.

Explanation:

Define unusual values in probability distributions

An unusual value in a probability distribution is typically defined as an outcome \(x\) with a probability of occurrence less than or equal to \(0.05\) (i.e., \(P(x) \le 0.05\)).

Analyze histogram (a) where \(n = 4\)

Looking at the first histogram for \(n = 4\):

• \(P(0) \le 0.05\) (the bar is below the \(0.05\) level)
• \(P(1) \approx 0.11 > 0.05\)
• \(P(2) \approx 0.31 > 0.05\)
• \(P(3) \approx 0.39 > 0.05\)
• \(P(4) \approx 0.19 > 0.05\)
• For \(x \ge 5\), the values are outside the range of possible trials since \(n = 4\).

Thus, the only unusual value is \(x = 0\).

Analyze histogram (b) where \(n = 8\)

Looking at the second histogram for \(n = 8\):

• \(P(0) \le 0.05\) (no visible bar, so \(P(0) \approx 0 \le 0.05\))
• \(P(1) \le 0.05\) (no visible bar, so \(P(1) \approx 0 \le 0.05\))
• \(P(2) \le 0.05\) (the bar is below the \(0.05\) level)
• \(P(3) \approx 0.08 > 0.05\)
• \(P(4) \approx 0.19 > 0.05\)
• \(P(5) \approx 0.29 > 0.05\)
• \(P(6) \approx 0.27 > 0.05\)
• \(P(7) \approx 0.14 > 0.05\)
• \(P(8) \le 0.05\) (the bar is below the \(0.05\) level)
• For \(x \ge 9\), the values are outside the range of possible trials since \(n = 8\).

Thus, the unusual values are \(x = 0\), \(x = 1\), \(x = 2\), and \(x = 8\).

Analyze histogram (c) where \(n = 12\)

Looking at the third histogram for \(n = 12\):

• \(P(0), P(1), P(2), P(3), P(4) \le 0.05\) (bars are either non-existent or below \(0.05\))
• \(P(5) \approx 0.06 > 0.05\)
• \(P(6) \approx 0.13 > 0.05\)
• \(P(7) \approx 0.21 > 0.05\)
• \(P(8) \approx 0.25 > 0.05\)
• \(P(9) \approx 0.20 > 0.05\)
• \(P(10) \approx 0.11 > 0.05\)
• \(P(11) \le 0.05\) (the bar is below the \(0.05\) level)
• \(P(12) \le 0.05\) (the bar is below the \(0.05\) level)

Thus, the unusual values are \(x = 0\), \(x = 1\), \(x = 2\), \(x = 3\), \(x = 4\), \(x = 11\), and \(x = 12\).

Answer:

Question 1

• A. \(x = 0, x = 1, x = 2\), and \(x = 8\)
• B. \(x = 0\) (Correct answer)
• C. There are no unusual values of x in the histogram.

Question 2

• A. \(x = 5\)
• B. \(x = 0\)
• C. \(x = 0, x = 1, x = 2\), and \(x = 8\) (Correct answer)
• D. There are no unusual values of x in the histogram.

Question 3

• A. \(x = 0, x = 1, x = 2, x = 3, x = 4, x = 11\), and \(x = 12\) (Correct answer)
• B. \(x = 0\)
• C. \(x = 0, x = 1, x = 2\), and \(x = 8\)
• D. There are no unusual values of x in the histogram.