Questions: According to an annual survey by Paytronix, around 17% of gift cards are still unused after a full year from purchase. Sounds like free money for businesses, right? It's all in the accounting - when gift cards are sold, they are counted as a liability until spent. A local restaurant chain sold 450 twenty-five dollar gift cards. Based on the data they expect that 83% of the cards will be used within a year of the purchase. Use the normal approximation along with the continuity correction factor to find the probability that at most 374 gift cards of 450 sold will be used within a year of purchase. - Let X be the number of gift cards out of 450 sold that will be used within a year from the purchase. Describe the distribution of X and its parameters: [ X-text Select an answer vee(n=square, p=square) ] - Use the random variable notation to symbolically express the probability that at most 374 gift cards out of 450 sold will be used within a year from the purchase: - 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=square, sigma=square) ] - Use the random variable notation to symbolically express the approximate probability that at most 374 gift cards out of 450 sold will be used within a year from the purchase: - Use the correction for continuity: - Find the probability (round the answer to 4 decimal places):

According to an annual survey by Paytronix, around 17% of gift cards are still unused after a full year from purchase. Sounds like free money for businesses, right? It's all in the accounting - when gift cards are sold, they are counted as a liability until spent. A local restaurant chain sold 450 twenty-five dollar gift cards. Based on the data they expect that 83% of the cards will be used within a year of the purchase. Use the normal approximation along with the continuity correction factor to find the probability that at most 374 gift cards of 450 sold will be used within a year of purchase. - Let X be the number of gift cards out of 450 sold that will be used within a year from the purchase. Describe the distribution of X and its parameters: [ X-text Select an answer vee(n=square, p=square) ] - Use the random variable notation to symbolically express the probability that at most 374 gift cards out of 450 sold will be used within a year from the purchase: - 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=square, sigma=square) ] - Use the random variable notation to symbolically express the approximate probability that at most 374 gift cards out of 450 sold will be used within a year from the purchase: - Use the correction for continuity: - Find the probability (round the answer to 4 decimal places):
Transcript text: According to an annual survey by Paytronix, around 17\% of gift cards are still unused after a full year from purchase. Sounds like free money for businesses, right? It's all in the accounting - when gift cards are sold, they are counted as a liability until spent. A local restaurant chain sold 450 twenty-five dollar gift cards. Based on the data they expect that $83 \%$ of the cards will be used within a year of the purchase. Use the normal approximation along with the continuity correction factor to find the probability that at most 374 gift cards of 450 sold will be used within a year of purchase. - Let $X$ be the number of gift cards out of 450 sold that will be used within a year from the purchase. Describe the distribution of $X$ and its parameters: \[ X-\text { Select an answer } \vee(n=\square, p=\square) \] - Use the random variable notation to symbolically express the probability that at most 374 gift cards out of 450 sold will be used within a year from the purchase: $\qquad$ - 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): \[ Y \sim \text { Select an answer } \vee(\mu=\square, \sigma=\square) \] $\qquad$ $\square$ - Use the random variable notation to symbolically express the approximate probability that at most 374 gift cards out of 450 sold will be used within a year from the purchase: $\qquad$ - Use the correction for continuity: $\qquad$ - 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 \( X \)

Let \( X \) be the number of gift cards out of 450 sold that will be used within a year from the purchase. The distribution of \( X \) can be described as: \[ X \sim \text{Binomial}(n=450, p=0.83) \]

Step 2: Probability Expression

We want to find the probability that at most 374 gift cards will be used within a year from the purchase. This can be expressed as: \[ P(X \leq 374) \]

Step 3: Parameters of the Normal Variable \( Y \)

To approximate the binomial distribution using a normal distribution, we calculate the mean \( \mu \) and standard deviation \( \sigma \): \[ \mu = n \cdot p = 450 \cdot 0.83 = 373.5 \] \[ \sigma^2 = n \cdot p \cdot q = 450 \cdot 0.83 \cdot (1 - 0.83) = 63.495 \] \[ \sigma = \sqrt{npq} = \sqrt{63.495} \approx 7.9684 \] Thus, the normal variable \( Y \) can be described as: \[ Y \sim \text{Normal}(\mu=373.5, \sigma=7.9684) \]

Step 4: Approximate Probability Expression

Using the continuity correction, we express the probability as: \[ P(Y \leq 374.5) \]

Step 5: Calculate the Probability

To find the probability, we calculate the Z-scores: \[ Z_{end} = \frac{374.5 - 373.5}{7.9684} \approx 0.1255 \] The probability can be expressed as: \[ P = \Phi(Z_{end}) - \Phi(-\infty) = \Phi(0.1255) - 0 = 0.5499 \]

Final Answer

The probability that at most 374 gift cards will be used is: \[ \boxed{0.5499} \]

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