Elena conducts an experiment in which she fills up the gas...

Elena conducts an experiment in which she fills up the gas tank on her Toyota Camry 40 times and records the miles per gallon for each fill-up. A histogram of the miles per gallon indicates that a normal distribution with mean of 24.6 miles per gallon and a standard deviation of 3.2 miles per gallon could be used to model the gas mileage for her car. (a) The following figure represents the normal curve with $\mu=24.6$ miles per gallon and $\sigma=3.2$ miles per gallon. The area under the curve to the right of $X=26$ is $0.3309 .$ Provide two interpretations of this area. (GRAPH CANNOT COPY) (b) The following figure represents the normal curve with $\mu=24.6$ miles per gallon and $\sigma=3.2$ miles per gallon. The area under the curve between $X=18$ and $X=21$ is 0.1107. Provide two interpretations of this area. (GRAPH CANNOT COPY)

Elena conducts an experiment in which she fills up the gas tank on her Toyota Camry 40 times and records the miles per gallon for each fill-up. A histogram of the miles per gallon indicates that a normal distribution with mean of 24.6 miles per gallon and a standard deviation of 3.2 miles per gallon could be used to model the gas mileage for her car.
(a) The following figure represents the normal curve with $\mu=24.6$ miles per gallon and $\sigma=3.2$ miles per gallon. The area under the curve to the right of $X=26$ is $0.3309 .$ Provide two interpretations of this area. (GRAPH CANNOT COPY)
(b) The following figure represents the normal curve with $\mu=24.6$ miles per gallon and $\sigma=3.2$ miles per gallon. The area under the curve between $X=18$ and $X=21$ is 0.1107. Provide two interpretations of this area. (GRAPH CANNOT COPY)

Key Concepts

Normal Distribution

The normal distribution is a probability distribution that is symmetric around its mean. It is fully described by its mean and standard deviation and is used to model real-valued random variables whose distributions are not known in detail but are assumed to follow this pattern due to the central limit theorem.

Area Under the Curve as Probability

In continuous probability distributions, the area under the curve over an interval represents the probability of the random variable falling within that interval. This interpretation applies universally, whether the area is in the tail of the distribution or between two specific values.

Tail Probabilities

Tail probabilities refer to the areas under the curve that lie beyond a certain cutoff point on one end of the distribution. They are used to quantify the likelihood of extreme outcomes, such as values higher than a particular threshold or lower than a specific limit.

Interval Probabilities

Interval probabilities are the probabilities that the random variable falls within a given range. These are determined by calculating the area under the curve between two particular values, providing insights into the frequency of outcomes occurring within that range.

Transcript

00:01 Welcome to numerid in the current problem we are given that elena conducts an experiment in which she fills up the gas tank on her toyota cam read 40 times and records the miles per gallon for each fill up.

00:14 Okay.

00:14 And the histogram is given which shows a normal distribution.

00:21 So x is the number of miles, okay? x is the number of miles per gallon, okay? and x follows a normal distribution with mean mu is equal to 24 .6 miles per gallon, i will be g.

00:55 And sigma is equal to 3 .2 mpg.

01:03 Now we have to first look at the diagram that they have provided and we have to, we are also given that if we have a normal distribution like this, it will be centered at 24 .6 and it will have a sigma distance which is 3 .2 more.

01:44 So that would be 3 .2 since i have 27 .8.

01:50 27 .8 and so on.

01:53 Now they have marked a line over here at 26 and shaded this region right side of it to be 0 .3309.

02:11 That is 33 .09%.

02:14 Now we have to give two interpretations.

02:18 So, the first interpretation goes like on an average, okay, on an in general, but each feeling out of 100 cases, approximately 33 % of the times it will exceed, it will exceed 26 miles per gallon.

03:03 Okay so that means each time there is a refill 33 times out of 100 it will exceed 26 and majorly or in general that is the remaining number of times 67 % of times it will be maximum 26 mpg that means it will remain to the left hand side okay so we are understanding how this number 26 is dividing the data into two different segments and what are the meaning of that what is the meaning of x less than equals to 26 and this is the case where x is greater than equals to 26 so i hope you could understand this let me know if you have any questions now after a plain screen uh let us again have the few basic details over here which is mu is equal to 24 .6 mpg and sigma is equals to 3 .2 mp.

04:32 Now we are given the normal curve which is centered as at 24 .6...

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