Questions: In the last decade, 63% of climbers have successfully climbed Mt. Everest on their first attempt. This year about 300 first-time climbers will attempt to climb the summit. Use the normal approximation along with complete the climb on their first attempt. - Let X be the number of climbers out of 300 that will successfully complete the climb. Describe the distribution of X and its parameters: - Use the random variable notation to symbolically express the probability that at least 183 first-time climbers will successfully complete the climb: - Let Y be a normal variable that will be used to approximate the probability in question. Find the parameters of Y (round the answers to 2 decimal places): - Use the random variable notation to symbolically express the approximate probability that at least 183 climbers will successfully complete the climb: - Use the correction for continuity: - Find the probability (round the answer to 4 decimal places):

In the last decade, 63% of climbers have successfully climbed Mt. Everest on their first attempt. This year about 300 first-time climbers will attempt to climb the summit. Use the normal approximation along with complete the climb on their first attempt. - Let X be the number of climbers out of 300 that will successfully complete the climb. Describe the distribution of X and its parameters: - Use the random variable notation to symbolically express the probability that at least 183 first-time climbers will successfully complete the climb: - Let Y be a normal variable that will be used to approximate the probability in question. Find the parameters of Y (round the answers to 2 decimal places): - Use the random variable notation to symbolically express the approximate probability that at least 183 climbers will successfully complete the climb: - Use the correction for continuity: - Find the probability (round the answer to 4 decimal places):
Transcript text: In the last decade, $63 \%$ of climbers have successfully climbed Mt. Everest on their first attempt. This year about 300 first-time climbers will attempt to climb the summit. Use the normal approximation along with complete the climb on their first attempt. - Let $X$ be the number of climbers out of 300 that will successfully complete the climb. Describe the distribution of $X$ and its parameters: \[ X \sim \text { Select an answer } \vee(n=\square, \] - Use the random variable notation to symbolically express the probability that at least 183 first-time climbers will successfully complete the climb: - Let $Y$ be a normal variable that will be used to approximate the probability in question. Find the parameters of $Y$ (round the answers to 2 decimal places): \[ Y \sim \text { Select an answer } \vee(\mu= \] , $\sigma=$ , - Use the random variable notation to symbolically express the approximate probability that at least 183 climbers will successfully complete the climb: - Use the correction for continuity: Select an answer - Find the probability (round the answer to 4 decimal places):
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Solution

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Solution Steps

Step 1: Distribution of \( X \)

The random variable \( X \), representing the number of climbers out of 300 that will successfully complete the climb on their first attempt, follows a binomial distribution given by:

\[ X \sim \text{Binomial}(n=300, p=0.63) \]

The parameters of this distribution are calculated as follows:

  • Mean \( \mu = n \cdot p = 300 \cdot 0.63 = 189.0 \)
  • Variance \( \sigma^2 = n \cdot p \cdot q = 300 \cdot 0.63 \cdot (1 - 0.63) = 69.93 \)
  • Standard Deviation \( \sigma = \sqrt{npq} = \sqrt{69.93} \approx 8.36 \)
Step 2: Probability Expression

We express the probability that at least 183 first-time climbers will successfully complete the climb as:

\[ P(X \geq 183) \]

Step 3: Normal Approximation Parameters

To approximate this probability using a normal distribution, we define the random variable \( Y \) as follows:

\[ Y \sim N(\mu=189.0, \sigma=8.36) \]

Step 4: Approximate Probability Expression

The approximate probability that at least 183 climbers will successfully complete the climb is expressed as:

\[ P(Y \geq 183) \]

Step 5: Continuity Correction

Applying the continuity correction, we adjust the probability to:

\[ P(Y \geq 182.5) \]

Step 6: Calculate the Probability

Using the standard normal distribution, we find the probability:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(-0.7775) \approx 0.7816 \]

Final Answer

The probability that at least 183 climbers will successfully complete the climb is:

\[ \boxed{0.7816} \]

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