The avaerage life span of bread-making machine is 7 years, with a standard deviation of 1 year. Assuming that the life span of these machines follows a normal distribution, find the probability that the mean life span of a random sample of 16 such machines falls between 6.6 and 7.2 years.
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with solution po sana and solving po. thank you need ko ASAP!
pag Mali auto report.
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Answer:
Approximately 0.7333, or 73.33%
Step-by-step explanation:
To solve this problem, we can use the central limit theorem and the standard normal distribution.
First, we need to calculate the standard error of the mean (SEM), which is the standard deviation of the population (1 year) divided by the square root of the sample size (16):
SEM = 1 / sqrt(16) = 0.25
Next, we can convert the range of 6.6 to 7.2 years into z-scores using the formula:
z = (x - μ) / SEM
where x is the value of the mean life span we're interested in, μ is the population mean (7 years), and SEM is the standard error of the mean we calculated earlier.
For the lower bound of 6.6 years:
z1 = (6.6 - 7) / 0.25 = -1.6
For the upper bound of 7.2 years:
z2 = (7.2 - 7) / 0.25 = 0.8
We can now look up the probabilities corresponding to these z-scores in the standard normal distribution table. The probability of z being less than -1.6 is 0.0548, and the probability of z being less than 0.8 is 0.7881. To find the probability of z being between these two values, we can subtract the smaller probability from the larger probability:
P(-1.6 < z < 0.8) = 0.7881 - 0.0548 = 0.7333
Therefore, the probability that the mean life span of a random sample of 16 bread-making machines falls between 6.6 and 7.2 years is approximately 0.7333, or 73.33%.