Suppose the life of refrigerators is normally distributed with mean μ=14 years and standard devi

Suppose the life of refrigerators is normally distributed...

Suppose the life of refrigerators is normally distributed with mean $\mu=14$ years and standard deviation $\sigma=2.5$ years. (Source: Based on information obtained from Consumer Reports) (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of refrigerators that are kept for more than 17 years. (c) Suppose the area under the normal curve to the right of $X=17$ is $0.1151 .$ Provide two interpretations of this result.

Suppose the life of refrigerators is normally distributed with mean $\mu=14$ years and standard deviation $\sigma=2.5$ years. (Source: Based on information obtained from Consumer Reports)
(a) Draw a normal curve with the parameters labeled.
(b) Shade the region that represents the proportion of refrigerators that are kept for more than 17 years.
(c) Suppose the area under the normal curve to the right of $X=17$ is $0.1151 .$ Provide two interpretations of this result.

Key Concepts

Graphical Representation of Normal Distribution

This involves drawing the bell-shaped curve of the normal distribution, labeling key parameters such as the mean and standard deviations, and shading specific regions under the curve to represent probabilities corresponding to intervals of interest.

Z-Score

The Z-score is a standardized value that indicates how many standard deviations a particular value is from the mean. It is used to convert a value from a normal distribution into a common scale, making it easier to calculate probabilities.

Interpretation of Areas Under the Normal Curve

The area under the normal curve represents probabilities. For example, shading the region to the right of a specific value, such as 17 years, corresponds to the proportion of cases that exceed this value. This interpretation is used to provide insights into statistical statements, such as the probability or percentage of occurrences beyond a given benchmark.

Mean

The mean is the central value of a distribution and indicates the average outcome. In a normal distribution, the mean determines the center point of the symmetrical curve where most values tend to cluster.

Normal Distribution

This is a continuous probability distribution characterized by a bell-shaped curve that is symmetric about its mean. It is used to model many natural phenomena, such as the life spans of products or measurement errors, where values near the mean are more common than those farther away.

Standard Deviation

Standard deviation measures the dispersion or spread of the data points around the mean. In a normally distributed dataset, it indicates how spread out the values are and helps determine the width of the bell curve.

Transcript

00:01 Welcome to numerid.

00:02 In the current problem we are given that life of refrigerators is the variable under consideration.

00:12 So life of, i will say, fridges.

00:16 It's easy and short than refrigerators.

00:19 And this follows a normal distribution with new is equals to 14 years and sigma is equal to 2 .5 years.

00:35 Now first thing that we are asked is to draw the curve, okay? and second, we have to find that the shield of the region that is less than equal to 17 years.

00:56 And the third one is that suppose this probability of x less than equals to 17 is equal to 0 .151 then what is a meaning of this? so for the diagram first itself we will get this over here and then continue to drop.

01:22 If you see the distribution has centered at 14 so this will be about 14.

01:29 So, so this is 14 and then 14 minus 2 .5 would be this because 14 minus 2 .5 that will be one sigma distance.

01:47 Correct.

01:48 Each of these are sigma, sigma, sigma, so it is minus 3 sigma and this is plus 3 sigma, right? so this would give us 11 .5.

02:02 Then again if i subtract 2 .5.

02:06 I get 9.

02:12 Again if i subtract 2 .5, i get 7 .5.

02:19 Now for 14, if i add 2 .5, i get 16 .5.

02:25 Again, if i add 2 .5, i will get 90.

02:31 If i again add, i will get 21 .5.

02:36 Now, if you see 17.

02:39 17 will lie somewhere over here, correct? so we have 17 over here.