For a certain eBay store, the prices of items can be approximated by a Normal model, with mean price $40 and standard deviation (SD) $5.

A customer has $30 to spend.

Approximately what percentage of the items could he buy?

Hint: Draw a sketch and justify your answer.

Question

For a certain eBay store, the prices of items can be approximated by a Normal model, with mean price $40 and standard deviation (SD) $5.

A customer has $30 to spend.

Approximately what percentage of the items could he buy?

Hint: Draw a sketch and justify your answer.

BlackTom AI · Accepted Answer

The z-score for $30 is (30 − 40)/5 = −2, so we need P(Z ≤ −2) for a standard normal distribution.

1. Relevant knowledge points and concepts
- Normal distribution properties: symmetric, described by mean (μ) and standard deviation (σ).
- Z-score transformation: z = (x − μ)/σ converts an x-value to the standard normal.
- Empirical/68–95–99.7 rule and standard normal cumulative probabilities: about 95% of values lie within ±2σ, leaving about 5% in the two tails combined, so about 2.5% in each tail. Exact table value for z = −2 is ≈ 0.0228 (2.28%).

2. Step-by-step reasoning
- Start with the model: X ~ N(μ = 40, σ = 5).
- Determine the cutoff: the customer can buy items priced ≤ $30.
- Convert $30 to a z-score: z = (30 − 40)/5 = −10/5 = −2.
- Compute the probability: P(X ≤ 30) = P(Z ≤ −2).

3. Calculations and intermediate steps
- z = −2.
- From a standard normal table or calculator: P(Z ≤ −2) = 0.0228 (approximately).
- Using the empirical rule as a quick check: about 95% lie between −2σ and +2σ, so roughly 2.5% lie below −2σ. This matches the exact value (2.28% ≈ 2.5%).

4. Why the correct answer is correct
- The z-score calculation shows $30 is two standard deviations below the mean. The standard normal cumulative probability at z = −2 gives the fraction of items priced at or below $30. Both the exact lookup (2.28%) and the empirical rule (≈2.5%) support the same conclusion.

5. Why other common answers would be incorrect
- 16% is wrong because it corresponds to one standard deviation below the mean (z = −1), not two.
- ≈0.15% would be wrong because it corresponds to z ≈ −3 (three standard deviations), not −2.
- Any answer near 50% or 84% confuses directions or uses the wrong side of the mean.

6. Conclusion
Therefore, approximately 2.5% of the items could be bought with $30, which matches the provided correct answer of 2.5.

For a certain eBay store, the prices of items can be approximated by a Normal model, with mean price $40 and standard deviation (SD) $5. A customer has $30 to spend. Approximately what percentage of the items could he buy? Hint: Draw a sketch and justify your answer.单项选择题

类似问题

A normal distribution is a bell-shaped curve with a majority of the cases falling within the Blank ______ of the range.

A bell-shaped probability curve demonstrates a(n) Blank ______ distribution.

The tensile strength of paper can be modelled by a normal distribution with a mean of 240kPa and standard deviation 15kPa. What proportion of paper produced will fail specifications, if the paper must have strength of at least 225kPa?

The tensile strength of paper can be modelled by a normal distribution with a mean of 240kPa and standard deviation 15kPa. What is the probability that the tensile strength will be between 226kPa and 243kPa?

Select the MOST appropriate statement for the population parameters of the Normal distribution shown below

The average length of a dog pregnancy is a normal random variable with mean 𝜇 = 63 days and standard deviation 𝜎 = 5 days. Use the empirical rule to determine what percent of dogs are pregnant longer than 68 days.

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单