# Normal Probability Distributions :

• Sixdaysgrace
In summary: so when you increase the mean to this new number, the fraction of people that live to this new number will be 70% of the people that lived to the old mean.
Sixdaysgrace
Hi Physics Forum! I hope this is the right section... I couldn't find a section on statistics.
This is a rather easy question but I can't seem to get the answer that the answer key says!

Average Life expectancy μ=72
standard Deviation ( in years ) δ=5

The question: Recent studies have suggested that advances in healthcare may increase the lifespan of employees. How much would the average life expectancy need to rise in order for 70% of people to live to age 70.

http://miha.ef.uni-lj.si/_dokumenti3plus2/195166/norm-tables.pdf
(Normal Distribution table in case you guys don't have !)

I don't know exactly how to approach this question. When it says X percent live up to Y age. Does that mean that the spread from the mean is X percent? Or up until Y age it is X percent.

If we try the former then that means 70 pecent of the empolyees live between the ages
66.85 and 77.15 ...but that doesn't seem like the right way to go because the answer has only one number.

If we try the latter that means 70 percent of the employees live from 0-74.6 (0.52 Z scores away from the mean)

But 74.6 is already higher than 70 which is what we are trying to get at...

I'm so lost can anybody help out?

Thanks

Sixdaysgrace said:
Hi Physics Forum! I hope this is the right section... I couldn't find a section on statistics.
This is a rather easy question but I can't seem to get the answer that the answer key says!

Average Life expectancy μ=72
standard Deviation ( in years ) δ=5

The question: Recent studies have suggested that advances in healthcare may increase the lifespan of employees. How much would the average life expectancy need to rise in order for 70% of people to live to age 70.

http://miha.ef.uni-lj.si/_dokumenti3plus2/195166/norm-tables.pdf
(Normal Distribution table in case you guys don't have !)

I don't know exactly how to approach this question. When it says X percent live up to Y age. Does that mean that the spread from the mean is X percent? Or up until Y age it is X percent.

If we try the former then that means 70 pecent of the empolyees live between the ages
66.85 and 77.15 ...but that doesn't seem like the right way to go because the answer has only one number.

If we try the latter that means 70 percent of the employees live from 0-74.6 (0.52 Z scores away from the mean)

But 74.6 is already higher than 70 which is what we are trying to get at...

I'm so lost can anybody help out?

Thanks

Under the current data, 70 is just 2/5 of a standard deviation below the mean (72), so the probability of a person surviving past 70 is $1 - \Phi(-2/5) \approx 0.655,$ a bit below the required 0.7. So, to what must you increase $\mu$ (the mean) in order to have $P\{ \text{Age} \geq 70 \} = 0.7$? Assume the standard deviation remains the same, since that is what the question implied.

RGV

Ray Vickson said:
Under the current data, 70 is just 2/5 of a standard deviation below the mean (72), so the probability of a person surviving past 70 is $1 - \Phi(-2/5) \approx 0.655,$ a bit below the required 0.7. So, to what must you increase $\mu$ (the mean) in order to have $P\{ \text{Age} \geq 70 \} = 0.7$? Assume the standard deviation remains the same, since that is what the question implied.

RGV

$1 - \Phi(-2/5) \approx 0.655,$

sir what is this equation? Could u please explain it for me?

1−Φ(−2/5)≈0.655,

Φ is the normal cdf function. Φ(x) gives you the area to the left of x in a normal distribution. since the whole area is 1, and Φ(-2.5) = .345, then the area to the right of -2/5 is .655.

earlier when you were got the 74. something years, that meant that 70% of people live 0 to 74.whatever years.

## What is a normal probability distribution?

A normal probability distribution, also known as a Gaussian distribution, is a type of probability distribution that follows a bell-shaped curve. It is used to describe the distribution of a continuous random variable where most of the data falls near the mean and the rest falls within a certain standard deviation.

## What are the characteristics of a normal probability distribution?

A normal distribution is characterized by its mean, median, and mode being equal. It also has a symmetrical bell-shaped curve, with 50% of the data falling on either side of the mean. The area under the curve is equal to 1, representing the total probability of all possible outcomes.

## How is a normal probability distribution calculated?

A normal distribution is calculated using the mean and standard deviation of a data set. The formula for calculating a normal distribution is (x-mean)/standard deviation. This formula is used to find the z-score, which represents the number of standard deviations a data point is from the mean. The z-score is then used to find the probability of a data point occurring within a certain range.

## What is the 68-95-99.7 rule for normal probability distributions?

The 68-95-99.7 rule, also known as the empirical rule, states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. This rule is useful for understanding the spread of data in a normal distribution.

## How is a normal probability distribution used in statistics?

A normal distribution is used in statistics to analyze and make predictions about data. It is commonly used in hypothesis testing and confidence interval calculations. It also allows us to make assumptions about the data and make inferences based on the characteristics of the distribution.

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