Question

Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally distributed, with a mean of 4.6 millimeters (mm) and a standard deviation of 1.0 mm. For a randomly found shard, find the following probabilities. (Round your answers to four decimal places.)

(a) the thickness is less than 3.0 mm

  

(b) the thickness is more than 7.0 mm

  

(c) the thickness is between 3.0 mm and 7.0 mm

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Quick Start Company makes 12-volt car batteries. After many years of product testing, the company knows that the average life of a Quick Start battery is normally distributed, with a mean of 46.6 months and a standard deviation of 8.5 months.

(a) If Quick Start guarantees a full refund on any battery that fails within the 36-month period after purchase, what percentage of its batteries will the company expect to replace? (Round your answer to two decimal places.)

%

(b) If Quick Start does not want to make refunds for more than 14% of its batteries under the full-refund guarantee policy, for how long should the company guarantee the batteries (to the nearest month)?

months

How much should a healthy kitten weigh? Suppose that a healthy 10-week-old (domestic) kitten should weigh an average of μ = 22.7 ounces with a (95% of data) range from 12.0 to 33.4 ounces. Let x be a random variable that represents the weight (in ounces) of a healthy 10-week-old kitten. Assume that x has a distribution that is approximately normal.

(a) The empirical rule (Section 7.1) indicates that for a symmetrical and bell-shaped distribution, approximately 95% of the data lies within two standard deviations of the mean. Therefore, a 95% range of data values extending from μ − 2σ to μ + 2σ is often used for "commonly occurring" data values. Note that the interval from μ − 2σ to μ + 2σ is 4σ in length. This leads to a "rule of thumb" for estimating the standard deviation from a 95% range of data values.

Estimating the standard deviation

For a symmetric, bell-shaped distribution,

standard deviation ≈

range

4

≈

high value − low value

4

where it is estimated that about 95% of the commonly occurring data values fall into this range.

Estimate the standard deviation of the x distribution. (Round your answer to two decimal places.)

oz

(b) What is the probability that a healthy 10-week-old kitten will weigh less than 14 ounces? (Round your answer to four decimal places.)

  

(c) What is the probability that a healthy 10-week-old kitten will weigh more than 33 ounces? (Round your answer to four decimal places.)

  

(d) What is the probability that a healthy 10-week-old kitten will weigh between 14 and 33 ounces? (Round your answer to four decimal places.)

  

(e) A kitten whose weight is in the bottom 9% of the probability distribution of weights is called undernourished. What is the cutoff point for the weight of an undernourished kitten? (Round your answer to two decimal places.)

oz

Answer 1

Given that

$\mu=4.6,\sigma=1$

(a)

The z-score for X = 3.0 is

$z=\frac{X-\mu}{\sigma}=\frac{3-4.6}{1}=-1.6$

Using z table, the probability that the thickness is less than 3.0 mm is

$P(X<3.0)=P(z<-1.6)=0.0548$

(b)

The z-score for X = 7.0 is

$z=\frac{X-\mu}{\sigma}=\frac{7-4.6}{1}=2.4$

Using z table, the probability that the thickness is more than 7.0 mm is

$P(X>7.0)=1-P(X\leq 7.0)=1-P(z\ \leq 2.4)=1-0.9918=0.0082$

(c)

Using z table, the probability that the thickness is between 3.0 mm and 7.0 mm is

$P(3\leq X\leq 7.0)=P(X\leq 7.0)-P(X\leq 3.0)=P(z\ \leq 2.4)-P(z\ \leq -1.6)=0.9918-0.0548=0.9370$