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Zorluk: OrtaNormal Distributions, Standard Deviation Curves, and Percentile Ranks

The continuous operating times of a model of industrial drone batteries are normally distributed with a mean of 220220 minutes and a standard deviation of 1515 minutes. A battery is designated as "high-efficiency" if its operating time places it in the top 16%16\% of all tested batteries. Based on the 68–95–99.768\text{--}95\text{--}99.7 empirical rule for normal distributions, what is the minimum operating time, in minutes, required for a battery to be designated as high-efficiency?

  1. A
    205205
  2. B
    220220
  3. 235235Cevap
  4. D
    236236
  5. E
    250250

Cevap

235235 minutes
According to the empirical rule for normal distributions, 68%68\% of all observations fall within 11 standard deviation of the mean (220±15220 \pm 15, or between 205205 and 235235). Because normal distributions are symmetric, the remaining 32%32\% of observations are split evenly between the upper and lower tails (16%16\% in each tail). The upper tail containing the top 16%16\% of battery operating times starts at 11 standard deviation above the mean, which is 220+15=235220 + 15 = 235 minutes.

Adım Adım Çözüm

1
Identify the given parameters of the normal distribution.
Mean μ=220\mu = 220 minutes and standard deviation σ=15\sigma = 15 minutes.
The problem specifies a normal distribution defined by these two parameters.
2
Apply the 68–95–99.768\text{--}95\text{--}99.7 empirical rule to determine the percentile threshold.
Approximately 68%68\% of the distribution falls within [μ−σ,μ+σ][\mu - \sigma, \mu + \sigma]. The unshaded area (100%−68%=32%100\% - 68\% = 32\%) is split symmetrically, with 16%16\% below μ−σ\mu - \sigma and 16%16\% above μ+σ\mu + \sigma.
By symmetry of the normal curve, the top 16%16\% corresponds precisely to values at or above 11 standard deviation above the mean (z=+1z = +1).
3
Calculate the raw score corresponding to z=+1z = +1.
Operating time =μ+1σ=220+1(15)=235= \mu + 1\sigma = 220 + 1(15) = 235 minutes.
Adding one standard deviation to the mean yields the minimum score required to be in the upper tail containing 16%16\% of the population.

Anahtar Kavram

Empirical Rule (68-95-99.7 Rule) and Symmetry of Normal Distributions
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Konular

Normal Distributions, Standard Deviation Curves, and Percentile Ranks

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