Questions: Market-share-analysis company Net Applications monitors and reports on Internet browser usage. According to Net Applications, in the summer of 2014, Google's Chrome browser exceeded a 20% market share for the first time, with a 20.37% share of the browser market (Forbes website). For a randomly selected group of 20 Internet browser users, answer the following questions. a. Compute the probability that exactly 8 of the 20 Internet browser users use Chrome as their Internet browser (to 4 decimals). b. Compute the probability that at least 3 of the 20 Internet browser users use Chrome as their Internet browser (to 4 decimals). c. For the sample of 20 Internet browser users, compute the expected number of Chrome users (to 3 decimals). d. For the sample of 20 Internet browser users, compute the variance and standard deviation for the number of Chrome users (to 3 decimals). Variance Standard deviation

Market-share-analysis company Net Applications monitors and reports on Internet browser usage. According to Net Applications, in the summer of 2014, Google's Chrome browser exceeded a 20% market share for the first time, with a 20.37% share of the browser market (Forbes website). For a randomly selected group of 20 Internet browser users, answer the following questions. a. Compute the probability that exactly 8 of the 20 Internet browser users use Chrome as their Internet browser (to 4 decimals). b. Compute the probability that at least 3 of the 20 Internet browser users use Chrome as their Internet browser (to 4 decimals). c. For the sample of 20 Internet browser users, compute the expected number of Chrome users (to 3 decimals). d. For the sample of 20 Internet browser users, compute the variance and standard deviation for the number of Chrome users (to 3 decimals). Variance Standard deviation
Transcript text: Market-share-analysis company Net Applications monitors and reports on Internet browser usage. According to Net Applications, in the summer of 2014, Google's Chrome browser exceeded a $20 \%$ market share for the first time, with a $20.37 \%$ share of the browser market (Forbes website). For a randomly selected group of 20 Internet browser users, answer the following questions. a. Compute the probability that exactly 8 of the 20 Internet browser users use Chrome as their Internet browser (to 4 decimals). For this question, if you compute the probability manually, make sure to carry at least six decimal digits in your calculations. $\square$ b. Compute the probability that at least 3 of the 20 Internet browser users use Chrome as their Internet browser (to 4 decimals). $\square$ c. For the sample of 20 Internet browser users, compute the expected number of Chrome users (to 3 decimals). $\square$ d. For the sample of 20 Internet browser users, compute the variance and standard deviation for the number of Chrome users (to 3 decimals). Variance $\square$ Standard deviation $\square$
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Solution

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Solution Steps

Hint

To solve these types of problems, use the binomial probability formula for calculating the likelihood of a specific number of successes in a series of independent trials. For cumulative probabilities, sum the probabilities of the desired range of outcomes or subtract the probabilities of the complement from one. The expected value and variance are found using the formulas for a binomial distribution, and the standard deviation is the square root of the variance.

Step 1: Compute the probability that exactly 8 of the 20 Internet browser users use Chrome

To find the probability that exactly 8 out of 20 users use Chrome, we use the binomial probability formula:

\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Given:

  • \( n = 20 \)
  • \( p = 0.2037 \)
  • \( k = 8 \)

The probability is:

\[ P(X = 8) \approx 0.0243 \]

Step 2: Compute the probability that at least 3 of the 20 Internet browser users use Chrome

To find the probability that at least 3 out of 20 users use Chrome, we calculate the cumulative probability for \( X \) being less than 3 and subtract it from 1:

\[ P(X \geq 3) = 1 - P(X < 3) \]

The probability is:

\[ P(X \geq 3) \approx 0.8051 \]

Step 3: Compute the expected number of Chrome users

The expected number of Chrome users in a sample of 20 is given by the formula for the expected value of a binomial distribution:

\[ E(X) = n \cdot p \]

Given:

  • \( n = 20 \)
  • \( p = 0.2037 \)

The expected value is:

\[ E(X) = 20 \cdot 0.2037 = 4.074 \]

Step 4: Compute the variance and standard deviation for the number of Chrome users

The variance of a binomial distribution is given by:

\[ \text{Var}(X) = n \cdot p \cdot (1 - p) \]

The standard deviation is the square root of the variance:

\[ \sigma = \sqrt{\text{Var}(X)} \]

Given:

  • \( n = 20 \)
  • \( p = 0.2037 \)

The variance is:

\[ \text{Var}(X) = 20 \cdot 0.2037 \cdot (1 - 0.2037) \approx 3.244 \]

The standard deviation is:

\[ \sigma \approx 1.801 \]

Final Answer

\[ \boxed{P(X = 8) \approx 0.0243} \] \[ \boxed{P(X \geq 3) \approx 0.8051} \] \[ \boxed{E(X) = 4.074} \] \[ \boxed{\text{Var}(X) \approx 3.244} \] \[ \boxed{\sigma \approx 1.801} \]

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