Probability (Normal Distribution etc.)

In summary, the conversation is about using the normal distribution equation to find the proportion or probability of bottles containing less than 2L of milk. The values for the normal distribution equation can be found in math tables, but the application of the equation may be challenging. The question asks for the probability of finding a bottle with less than 2L of milk, which is equivalent to finding the area under the curve for P(X<2). To use the tables, one must convert the normal distribution to a standard normal distribution with mean 0 and standard deviation 1.
  • #1
Master J
226
0
A factory makes milk that is sold as 2 L bottles. The amount of milk per bottle obeys a normal distribution with mean 2.02 and standard deviation 0.09. What proportion of bottles have less than 2 L in them?

Now the exact interpretation etc. of the normal distribution equation is beyond my course at this point, but it, and the values for it, are tabulated in my set of math tables. But I am struggling to understand how to apply it.
Could someone run thru its application in this question?

Thank you very much!
 
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  • #2
proportion is synonymous with probability. The question is asking you to find the probability of finding a bottle with less than 2L of milk in it, i.e. P(X<2).

Does that help you get started?
 
  • #3
Do you know how to change from a normal distribution with mean [itex]\mu[/itex] and standard deviation [itex]\sigma[/itex] to the "standard normal distribution" (mean 0, standard deviation 1) which is what the tables give?
 

1. What is the normal distribution?

The normal distribution is a type of probability distribution that is commonly used in statistics. It is a bell-shaped curve that is symmetrical and has a mean, median, and mode that are all equal. It is often referred to as a bell curve because of its shape.

2. How is the normal distribution calculated?

The normal distribution is calculated using a mathematical formula that takes into account the mean and standard deviation of a dataset. The formula is:
P(x) = (1 / (σ * √(2π))) * e^((-1/2) * ((x-μ)/σ)^2)
where P(x) is the probability of a given value, σ is the standard deviation, μ is the mean, and e is the base of natural logarithms.

3. What is the use of the normal distribution in real life?

The normal distribution is used in many real-life scenarios, such as in quality control, finance, and social sciences. It is particularly useful in predicting outcomes and understanding the likelihood of certain events occurring. For example, it can be used to determine the probability of a certain stock price or the height of a group of people falling within a certain range.

4. How is the normal distribution related to the central limit theorem?

The central limit theorem states that when a large enough sample size is taken from any population, the distribution of the sample means will be approximately normally distributed, regardless of the shape of the population's distribution. This means that the normal distribution is a fundamental concept in statistics, as it allows for the analysis and interpretation of data from various populations.

5. How can the normal distribution be used in hypothesis testing?

In hypothesis testing, the normal distribution is used to determine the probability of a certain outcome occurring by chance. The area under the curve represents the probability, and using critical values and the z-score, we can determine if a result is statistically significant or not. This is often used in medical research, where the effects of a certain treatment are compared to a control group.

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