Questions: What should I buy? A study conducted by a research group in a recent year reported that 38% of cell phone owners used their phones inside a store for guidance on purchasing decisions. A sample of 16 cell phone owners is studied. Round the answers to at least four decimal places. Part 1 of 4 (a) What is the probability that six or more of them used their phones for guidance on purchasing decisions? The probability that six or more of them used their phones for guidance on purchasing decisions is [ ].

What should I buy? A study conducted by a research group in a recent year reported that 38% of cell phone owners used their phones inside a store for guidance on purchasing decisions. A sample of 16 cell phone owners is studied. Round the answers to at least four decimal places. Part 1 of 4 (a) What is the probability that six or more of them used their phones for guidance on purchasing decisions? The probability that six or more of them used their phones for guidance on purchasing decisions is [ ].
Transcript text: What should I buy? A study conducted by a research group in a recent year reported that 38% of cell phone owners used their phones inside a store for guidance on purchasing decisions. A sample of 16 cell phone owners is studied. Round the answers to at least four decimal places. Part 1 of 4 (a) What is the probability that six or more of them used their phones for guidance on purchasing decisions? The probability that six or more of them used their phones for guidance on purchasing decisions is [ ].
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Solution

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Solution Steps

Step 1: Define the Problem

We are tasked with finding the probability that six or more out of 16 cell phone owners used their phones for guidance on purchasing decisions, given that the probability of a cell phone owner using their phone in this manner is \( p = 0.38 \).

Step 2: Calculate Individual Probabilities

We will calculate the probability of exactly \( x \) successes for \( x = 6, 7, \ldots, 16 \) using the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

where \( n = 16 \) and \( q = 1 - p = 0.62 \).

The calculated probabilities for each \( x \) are as follows:

  • \( P(X = 6) = 0.2024 \)
  • \( P(X = 7) = 0.1772 \)
  • \( P(X = 8) = 0.1222 \)
  • \( P(X = 9) = 0.0666 \)
  • \( P(X = 10) = 0.0286 \)
  • \( P(X = 11) = 0.0095 \)
  • \( P(X = 12) = 0.0024 \)
  • \( P(X = 13) = 0.0005 \)
  • \( P(X = 14) = 0.0001 \)
  • \( P(X = 15) = 0.0 \)
  • \( P(X = 16) = 0.0 \)
Step 3: Sum the Probabilities

To find the probability of six or more successes, we sum the probabilities from \( x = 6 \) to \( x = 16 \):

\[ P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15) + P(X = 16) \]

Calculating this gives:

\[ P(X \geq 6) = 0.2024 + 0.1772 + 0.1222 + 0.0666 + 0.0286 + 0.0095 + 0.0024 + 0.0005 + 0.0001 + 0.0 + 0.0 = 0.6095 \]

Final Answer

The probability that six or more of the cell phone owners used their phones for guidance on purchasing decisions is

\[ \boxed{0.6095} \]

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