Refrigerator Replacement Consumer Reports indicated that the...

Refrigerator Replacement Consumer Reports indicated that the average life of a refrigerator before replacement is $\mu=14$ years with a $(95 \% \text { of data) range from } 9 \text { to } 19 \text { years. Let } x=$ age at which a refrigerator is replaced. Assume that $x$ has a distribution that is approximately normal. (a) Find a good approximation for the standard deviation of $x$ values. Hint: See Problem 31. (b) What is the probability that someone will keep a refrigerator fewer than 11 years before replacement? (c) What is the probability that someone will keep a refrigerator more than 18 years before replacement? (d) Inverse Normal Distribution Assume that the average life of a refrigerator is 14 years, with the standard deviation given in part (a) before it breaks. Suppose that a company guarantees refrigerators and will replace a refrigerator that breaks while under guarantee with a new one. However, the company does not want to replace more than $5 \%$ of the refrigerators under guarantee. For how long should the guarantee be made (rounded to the nearest tenth of a year)?

Refrigerator Replacement Consumer Reports indicated that the average life of a refrigerator before replacement is $\mu=14$ years with a $(95 \% \text { of data) range from } 9 \text { to } 19 \text { years. Let } x=$ age at which a refrigerator is replaced. Assume that $x$ has a distribution that is approximately normal.
(a) Find a good approximation for the standard deviation of $x$ values. Hint:
See Problem 31.
(b) What is the probability that someone will keep a refrigerator fewer than
11 years before replacement?
(c) What is the probability that someone will keep a refrigerator more than 18 years before replacement?
(d) Inverse Normal Distribution Assume that the average life of a refrigerator is 14 years, with the standard deviation given in part (a) before it breaks. Suppose that a company guarantees refrigerators and will replace a refrigerator that breaks while under guarantee with a new one. However, the company does not want to replace more than $5 \%$ of the refrigerators under guarantee. For how long should the guarantee be made (rounded to the nearest tenth of a year)?

Key Concepts

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve, and defined by two parameters: the mean (which determines the center of the distribution) and the standard deviation (which measures the spread or dispersion). It is widely used in statistics because many real-world variables tend to follow this distribution, especially when influenced by many small, random factors.

Standard Deviation and the Empirical Rule

The standard deviation is a measure of how spread out the values of a distribution are around the mean. The empirical rule, or the 68-95-99.7 rule, states that approximately 95% of the data in a normal distribution lie within two standard deviations of the mean. This property can be used to approximate the standard deviation when given a range that covers 95% of the data.

Z-Scores

A z-score represents the number of standard deviations a data point is from the mean of the distribution. By converting raw scores to z-scores, one can standardize different normal distributions to the standard normal distribution, allowing for the use of standard normal tables or computer functions to find probabilities associated with specific outcomes.

Inverse Normal Distribution

The inverse normal distribution, or quantile function, is used to determine the value (or threshold) below or above which a certain percentage of data falls in a normal distribution. This is particularly useful in decision-making contexts where one needs to set guarantees or limits based on specified probabilities, such as ensuring that only a certain percentage of outcomes are considered acceptable.

Transcript

0:00 32.

00:02 Replacement consumer reports indicate that the average life of a refrigerator before replacement is 14 years.

00:08 We're going to write that down with a range of 9 to 19 years.

00:15 Let x equal the age at which a refrigerator is replaced.

00:18 Assume that x has a distribution that is approximately normal.

00:25 A, find a good approximation for the standard deviation of x values.

00:34 So for a, we want to find a good approximation for the standard deviation for the standard deviation.

00:38 Well, in the original part of the problem, we were told that we had 95 % of the data gives a range from 9 to 19.

00:47 Well, this allows us to see essentially on a curve that we have 9 to 19.

00:59 We know that our mean is 14.

01:01 So what we're going to do here is we're going to figure out what standard deviation would stretch out two to the right and two to the left.

01:12 To get us these boundary values of 9 and 19.

01:15 Now we're doing two because of 95%.

01:18 We know that empirical rule says approximately 95 % of the data falls within two standard deviations.

01:24 So we're just going to look at one side of this because it's symmetrical.

01:27 We have a difference of five.

01:29 That's supposed to span two standard deviations.

01:31 So each standard deviation must be approximately two and a half.

01:42 So we're going to use 2 .5 as our estimate for our standard deviation.

01:49 What is the probability that someone would keep a refrigerator fewer than 11 years before replacement? the probability that someone would keep a refrigerator fewer than 11 years.

01:59 So we want less than 11.

02:01 So on this same curve, i'll just draw a little neater version of it, where we have 14 in the center, 16 .5, 19, and there's a third we can put, but we're not going to worry about that.

02:21 I want to know the probability of less than 11, which is somewhere in the center.

02:25 Where we want to find this green region.

02:30 Well, you can use our ti 83 or 84 calculator under the normal cdf command.

02:37 Normal cdf allows us to find this area.

02:39 We just need to know where it starts, where it stops, what the mean, and what the standard deviation is.

02:44 And i'll show you what these letters are.

02:45 So l is lower bound.

02:47 Where does the shading start? technically, it starts at negative infinity because the curve never touches the axis...

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