A manufacturer has developed a new fishing line, which the company claims has a mean breaking strength of 15 kilograms with a standard deviation of 0.5 kilogram. To test the hypothesis that µ = 15 kilograms against the alternative that µ < 15 kilograms, a random sample of 50 lines will be tested. The critical region is defined to be xbar < 14.9. a) Find the probability of committing a type I error when H_0 is true.

Using the normal distribution and the central limit theorem, it is found that there is a 0.0793 = 7.93% probability of committing a type I error when [tex]H_0[/tex] is true.

In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

It measures how many standard deviations the measure is from the mean.
After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.
By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

In this problem:

Mean of 15 kg, hence [tex]\mu = 15[/tex]
Standard deviation of 0.5 kg, hence [tex]\sigma = 0.5[/tex].
Sample of 50, hence [tex]n = 50, s = \frac{0.5}{\sqrt{50}}[/tex]

A type I error is finding a sample mean below 14.9 kg, hence, the probability is the p-value of Z when X = 14.9.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem:

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{14.9 - 15}{\frac{0.5}{\sqrt{50}}}[/tex]

[tex]Z = -1.41[/tex]

[tex]Z = -1.41[/tex] has a p-value of 0.0793.

0.0793 = 7.93% probability of committing a type I error when [tex]H_0[/tex] is true.

A similar problem is given at https://brainly.com/question/24663213