Question

Industrial light bulbs should have a mean life length acceptable to potential users and a relatively small variation in life length. If some bulbs fail too early in their life, users become annoyed and are likely to switch to bulbs produced by a different manufacturer. Large variations above the mean reduce replacement sales; in general, variation in life lengths disrupts the user's replacement schedules. A random sample of 20 bulbs produced by a particular manufacturer produced the following lengths of life (in hours): 2100, 2302, 1951, 2067, 2415, 1883, 2101, 2146, 2278, 2019, 1924, 2183, 2077, 2392, 2286, 2501, 1946, 2161, 2253, 1827. (a) Find a 99.

Answer 1

The probability problems involve exponential distributions and require understanding of its density function, while the hypothesis testing problems assess manufacturing claims using the sample mean and known standard deviation.

Step-by-step explanation:

The question revolves around the statistics topic within mathematics, focusing on probabilistic models and hypothesis testing to assess the life span of products such as light bulbs and batteries. Students need to understand the exponential distribution to solve problems related to the lifespan of light bulbs and use the properties of the normal distribution for hypothesis testing to determine the reasonableness of product lifespan claims.

For an exponential distribution with a mean lifespan of eight years, calculations would involve the exponential probability density function to find the likelihood of a bulb failing within a specific time period. To assess the claims of a manufacturer regarding the lifespan of a product, like the batteries or tires, students would need to understand and apply hypothesis testing using the sample mean and known standard deviations to evaluate whether the observed data significantly deviates from the claimed parameters.