Answer :
Answer:
a
[tex]P(X < 12) = 0.3660[/tex]
b
[tex]P(X < 12) = 0.1521[/tex]
Step-by-step explanation:
Considering question a
From the question we are told that
The mean is [tex]\mu = 12.01 \ oz[/tex]
The standard deviation is [tex]\sigma = 0.35 \ oz[/tex]
The standard error of mean is mathematically represented as
[tex]\sigma_{x} = \frac{\sigma}{\sqrt{n} }[/tex]
=> [tex]\sigma_{x} = \frac{0.35}{\sqrt{144} }[/tex]
=> [tex]\sigma_{x} = 0.02917[/tex]
Generally the probability that the mean volume of a random sample of 144 bottles is less than 12 oz is mathematically represented as
[tex]P(X < 12) = P(\frac{X - \mu }{\sigma_{x}} < \frac{12 - 12.01 }{0.02917 } )[/tex]
[tex]\frac{X -\mu}{\sigma_{x} } = Z (The \ standardized \ value\ of \ X )[/tex]
[tex]P(X < 12) = P(Z < -0.3425 )[/tex]
From the z table the area under the normal curve to the left corresponding to -0.3425 is
[tex]P(X < 12) = P(Z < -0.3425 ) = 0.3660[/tex]
=> [tex]P(X < 12) = 0.3660[/tex]
Considering question a
From the question we are told that
The mean is [tex]\mu = 12.03 \ oz[/tex]
Generally the probability that the mean volume of a random sample of 144 bottles is less than 12 oz is mathematically represented as
[tex]P(X < 12) = P(\frac{X - \mu }{\sigma_{x}} < \frac{12 - 12.03 }{0.02917 } )[/tex]
[tex]P(X < 12) = P(Z < -1.0274 )[/tex]
From the z table the area under the normal curve to the left corresponding to -1.0274 is
[tex]P(X < 12) = P(Z < -1.0274 ) =0.1521[/tex]
=> [tex]P(X < 12) = 0.1521[/tex]
Final answer:
To find the probability that the mean volume of a sample is less than a certain value, we can use the Central Limit Theorem and the z-score formula. For a sample size of 144 bottles, the probability that the mean volume is less than 12 oz is 4.3%. However, if the population mean fill volume is increased to 12.03 oz, the probability increases to 19.4%.
Explanation:
In order to find the probability that the mean volume of a random sample of 144 bottles is less than 12 oz, we need to use the Central Limit Theorem. The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as the sample size increases. To find the probability, we can use the z-score formula and the standard normal distribution table.
a) The z-score is calculated using the formula z = (x - μ) / (σ / √n), where x is the sample mean, μ is the population mean, σ is the standard deviation, and n is the sample size. Plugging in the values, we get z = (12 - 12.01) / (0.35 / √144) = -1.714. Looking up the z-score in the standard normal distribution table, we find that the probability is approximately 0.043, or 4.3%.
b) If the population mean fill volume is increased to 12.03 oz, we need to recalculate the z-score. Using the same formula, we get z = (12 - 12.03) / (0.35 / √144) = -0.857. Looking up the z-score in the standard normal distribution table, we find that the probability is approximately 0.194, or 19.4%.
Learn more about Probability and Central Limit Theorem here:
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