The problem states that the prices paid for a particular OLED television are normally distributed with a mean of $1500 and a standard deviation of $95. We will use the standard normal distribution (Z-distribution) to calculate the probabilities for each of the scenarios. For this, we will convert the prices into Z-scores using the formula:

$Z = \frac{X - \mu}{\sigma}$

Where:

• $X$ is the price,
• $\mu = 1500$ (mean),
• $\sigma = 95$ (standard deviation).

(a) Probability that a randomly selected buyer paid more than $1700:

1. First, calculate the Z-score for $X = 1700$:

$Z = \frac{1700 - 1500}{95} = \frac{200}{95} \approx 2.1053$

2. Using a standard normal distribution table or a calculator, find the probability that $Z > 2.1053$. The area to the left of $Z = 2.1053$ is approximately 0.9824. Therefore, the probability of paying more than $1700 is:

$P(X > 1700) = 1 - 0.9824 = 0.0176$

So, the probability is approximately 0.0176.

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(b) Probability that a randomly selected buyer paid between $1400 and $1600:

1. Calculate the Z-score for $X = 1400$:

$Z = \frac{1400 - 1500}{95} = \frac{-100}{95} \approx -1.0526$

2. Calculate the Z-score for $X = 1600$:

$Z = \frac{1600 - 1500}{95} = \frac{100}{95} \approx 1.0526$

3. Find the probabilities corresponding to these Z-scores:

   • For $Z = -1.0526$, the area to the left is approximately 0.1462.
   • For $Z = 1.0526$, the area to the left is approximately 0.8538.

4. The probability that a buyer paid between $1400 and $1600 is the difference between these two probabilities:

$P(1400 < X < 1600) = 0.8538 - 0.1462 = 0.7076$

So, the probability is approximately 0.7076.

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(c) Probability that a randomly selected buyer paid less than $1350:

1. Calculate the Z-score for $X = 1350$:

$Z = \frac{1350 - 1500}{95} = \frac{-150}{95} \approx -1.5789$

2. Using a standard normal distribution table or a calculator, find the probability corresponding to $Z = -1.5789$. The area to the left of this Z-score is approximately 0.0572.

Therefore, the probability that a buyer paid less than $1350 is approximately 0.0572.

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Summary of Answers:

• (a) The probability that a randomly selected buyer paid more than $1700 is 0.0176.
• (b) The probability that a randomly selected buyer paid between $1400 and $1600 is 0.7076.
• (c) The probability that a randomly selected buyer paid less than $1350 is 0.0572.

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Would you like more details or have any questions on this?

Related Questions:

1. What is the probability that a buyer paid between $1300 and $1700?
2. What is the price at which 90% of the buyers paid more?
3. What is the probability that a buyer paid exactly the mean price?
4. If you were to randomly select 10 buyers, what is the probability that exactly one of them paid more than $1700?
5. What price would correspond to the 95th percentile of the distribution?

Tip:

In normal distribution problems, converting raw scores to Z-scores allows you to use standard tables or calculators for probabilities, simplifying the process!