Questions: What should I buy? A study conducted by a research group in a recent year reported that 56% of cell phone owners used their phones inside a store for guidance on purchasing decisions. A sample of 14 cell phone owners is studied. Round the answers to at least four decimal places. Part 2 of 4 (b) What is the probability that fewer than nine of them used their phones for guidance on purchasing decisions?

What should I buy? A study conducted by a research group in a recent year reported that 56% of cell phone owners used their phones inside a store for guidance on purchasing decisions. A sample of 14 cell phone owners is studied. Round the answers to at least four decimal places. Part 2 of 4 (b) What is the probability that fewer than nine of them used their phones for guidance on purchasing decisions?
Transcript text: What should I buy? A study conducted by a research group in a recent year reported that $56 \%$ of cell phone owners used their phones inside a store for guidance on purchasing decisions. A sample of 14 cell phone owners is studied. Round the answers to at least four decimal places. Part 1 of 4 (a) What is the probability that five or more of them used their phones for guidance on purchasing decisions? The probability that five or more of them used their phones for guidance on purchasing decisions is 0.9639 . Part: $1 / 4$ Part 2 of 4 (b) What is the probability that fewer than nine of them used their phones for guidance on purchasing decisions?
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Solution

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Solution Steps

Step 1: Define the Problem

We are analyzing the usage of cell phones by owners inside a store for guidance on purchasing decisions. The probability of a cell phone owner using their phone for this purpose is \( p = 0.56 \). We are studying a sample of \( n = 14 \) cell phone owners.

Step 2: Calculate Probability for Part (a)

We need to find the probability that five or more cell phone owners used their phones for guidance. This can be expressed mathematically as:

\[ P(X \geq 5) = 1 - P(X < 5) = 1 - \sum_{x=0}^{4} P(X = x) \]

Calculating \( P(X = x) \) for \( x = 0, 1, 2, 3, 4 \):

  • \( P(X = 0) = \binom{14}{0} \cdot (0.56)^0 \cdot (0.44)^{14} = 0.0003 \)
  • \( P(X = 1) = \binom{14}{1} \cdot (0.56)^1 \cdot (0.44)^{13} = 0.0002 \)
  • \( P(X = 2) = \binom{14}{2} \cdot (0.56)^2 \cdot (0.44)^{12} = 0.0015 \)
  • \( P(X = 3) = \binom{14}{3} \cdot (0.56)^3 \cdot (0.44)^{11} = 0.0076 \)
  • \( P(X = 4) = \binom{14}{4} \cdot (0.56)^4 \cdot (0.44)^{10} = 0.0268 \)

Summing these probabilities:

\[ P(X < 5) = 0.0003 + 0.0002 + 0.0015 + 0.0076 + 0.0268 = 0.0364 \]

Thus, the probability that five or more used their phones is:

\[ P(X \geq 5) = 1 - 0.0364 = 0.9640 \]

Step 3: Calculate Probability for Part (b)

Next, we need to find the probability that fewer than nine cell phone owners used their phones. This can be expressed as:

\[ P(X < 9) = \sum_{x=0}^{8} P(X = x) \]

Calculating \( P(X = x) \) for \( x = 0, 1, 2, \ldots, 8 \):

  • \( P(X = 0) = 0.0003 \)
  • \( P(X = 1) = 0.0002 \)
  • \( P(X = 2) = 0.0015 \)
  • \( P(X = 3) = 0.0076 \)
  • \( P(X = 4) = 0.0268 \)
  • \( P(X = 5) = 0.0682 \)
  • \( P(X = 6) = 0.1301 \)
  • \( P(X = 7) = 0.1892 \)
  • \( P(X = 8) = 0.2108 \)

Summing these probabilities:

\[ P(X < 9) = 0.0003 + 0.0002 + 0.0015 + 0.0076 + 0.0268 + 0.0682 + 0.1301 + 0.1892 + 0.2108 = 0.6344 \]

Final Answer

The probabilities are as follows:

  • The probability that five or more used their phones: \( \boxed{0.9640} \)
  • The probability that fewer than nine used their phones: \( \boxed{0.6344} \)
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