Binomial Distribution for Manufacturer's Claim on Product Durability

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In summary, the conversation discusses a question about the distribution type for a manufacturer's claim that at most 5% of their product will require service before 1000 hours of operation. It is determined that the distribution is binomial and the probability of 3 products requiring service is 0.0754. The confusion arises from the wording of the claim, but it is concluded that the manufacturer's claim is accurate.
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dfraser
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Hi,

I'm struggling to know what distribution this question requires, and what should be signalling the distribution type:

A manufacturer claims at most 5% of his product will sustain fewer than 1000hrs of operation before needing service. Twenty products are selected at random from the production line and tested. It was found that 3 of them required service before 1000hrs of operation. Comment on the manufacturers claim.

The solution for the probability of 3 requiring service before the 1000hrs is (according to my text) 0.0754.

I think I might be getting confused by "at most 5% of his product will sustain fewer than 1000hrs".

For a binomial I'd have:
$${ 20\choose17 }{.95}^{17}{.05}^{3} = .05958$$

For a negative binomial I'd have:
$${ 19\choose2 }{.95}^{17}{.05}^{3} = .00893$$

This isn't poisson, geometric, hypergeometric, or multinomial, so what is it? Or can someone give me a hint about where I'm going wrong?
 
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  • #2
dfraser said:
Hi,

I'm struggling to know what distribution this question requires, and what should be signalling the distribution type:

A manufacturer claims at most 5% of his product will sustain fewer than 1000hrs of operation before needing service. Twenty products are selected at random from the production line and tested. It was found that 3 of them required service before 1000hrs of operation. Comment on the manufacturers claim.

The solution for the probability of 3 requiring service before the 1000hrs is (according to my text) 0.0754.

I think I might be getting confused by "at most 5% of his product will sustain fewer than 1000hrs".

For a binomial I'd have:
$${ 20\choose17 }{.95}^{17}{.05}^{3} = .05958$$

For a negative binomial I'd have:
$${ 19\choose2 }{.95}^{17}{.05}^{3} = .00893$$

This isn't poisson, geometric, hypergeometric, or multinomial, so what is it? Or can someone give me a hint about where I'm going wrong?

Hi dfraser,

It's a binomial distribution as you surmised.
It's just that they ask for $P(X\ge 3)$ instead of $P(X=3)$, where $X$ is the number that requires service before 1000 hrs.
And indeed we assume that the probability is 5% per product, since this gives the manufacturer the most leeway.
 

Related to Binomial Distribution for Manufacturer's Claim on Product Durability

What distribution is it?

The distribution of a dataset refers to the way the data points are spread out or distributed. It can provide valuable information about the central tendency, variability, and shape of the data.

1. What is a normal distribution?

A normal distribution, also known as a Gaussian distribution, is a bell-shaped distribution that is symmetrical around the mean. In a normal distribution, the majority of the data points are clustered around the mean, with a smaller number of points in the tails. Many natural phenomena, such as height and weight, follow a normal distribution.

2. What is a skewed distribution?

A skewed distribution is a distribution in which the data is not symmetrical around the mean. This means that the majority of the data points are clustered on one side of the distribution, causing the distribution to be skewed to one side. There are two types of skewed distributions: positively skewed (skewed to the right) and negatively skewed (skewed to the left).

3. How do you determine the type of distribution?

The type of distribution can be determined by looking at the shape of the data and calculating measures of central tendency and variability. A visual inspection of a histogram or a box plot can also help identify the type of distribution. Additionally, statistical tests such as the Shapiro-Wilk test can be used to determine if the data follows a normal distribution.

4. What is a uniform distribution?

A uniform distribution is a distribution in which all data points have an equal probability of occurring. In a uniform distribution, the data is evenly spread out across the range of values, resulting in a rectangular shape on a histogram. A classic example of a uniform distribution is rolling a fair die, where each number has an equal chance of being rolled.

5. Can a dataset have more than one type of distribution?

Yes, a dataset can have more than one type of distribution. This is known as a multimodal distribution, where there are multiple peaks or clusters in the data. A dataset can also have a combination of both continuous and discrete distributions, depending on the nature of the data. It is important to identify and understand the different types of distributions present in a dataset to accurately analyze and interpret the data.

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