Hi,

Below please find the solution, including explanation of an alternative approach. The previous approach has some issues and typos. Here is a revised simpler approach.

A tip is appreciated.

**Solution**

The beta life of a product with a Weibull distribution is the time at which a certain percentage of the products will have failed.

For example, if the beta life is 400 hours, it means that at 400 hours, a certain percentage of the products will have failed.

The slope of the Weibull distribution, also known as the shape parameter, determines the shape of the distribution. A slope of 1.66 indicates that the distribution is relatively steep, which means that the percentage of products that are expected to have failed at 1,600 hours will be relatively high.

To determine how long the redesigned products must be tested, you will need to use the inverse Weibull distribution function to solve for the time required to achieve a certain confidence level.

The inverse Weibull distribution function is given by:

t = (-ln(1 - confidence level))^(1/shape parameter) * beta life

Plugging in the values given in the question, we have:

t = (-ln(1 - 0.95))^(1/1.66) * 1600 hours

Solving this equation gives us a value of approximately 3,100 hours. This means that the redesigned products must be tested for at least 3,100 hours without failure in order to achieve a confidence level of 95% that they have a beta life of at least 1,600 hours.

It's worth noting that this is just a rough estimate and the actual testing time may vary depending on the specifics of the product and the testing conditions.

**Alternative solution approach**

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Alternatively, to determine how long the redesigned products must be tested to meet the goal of 95% confidence level, we need to know the percentage of products that are expected to have failed at the desired beta life of 1,600 hours.

After calculating that, in order to meet the goal of 95% confidence level, we need to test the redesigned products for a longer period of time until the percentage of products that have failed reaches 95%.

This will give the time the redesigned products must be tested for without failure to meet the goal of 95% confidence level.