The amount of water in a bottle is approximately normally distributed with a mean of 2.50 liters with a standard deviation of 0.025 liter. Complete parts (a) through (e) below. a. What is the probability that an individual bottle contains less than 2.48 liters? 0.212 (Round to three decimal places as needed.) b. If a sample of 4 bottles is selected, what is the probability that the sample mean amount contained is less than 2.48 liters? 0.055 (Round to three decimal places as needed.) c. If a sample of 25 bottles is selected, what is the probability that the sample mean amount contained is less than 2.48 liters? 0.000 (Round to three decimal places as needed.) d. Explain the difference in the results of (a) and (c). Part (a) refers to an individual bottle, which can be thought of as a sample with sample size 2.48|. Therefore, the standard error of the mean for an individual bottle is 0 times the standard error of the sample in (c) with sample size 25. This leads to a probability in part (a) that is larger than the probability in part (c). (Type integers or decimals. Do not round.) e. Explain the difference in the results of (b) and (c). The sample size in (c) is greater than the sample size in (b), so the standard error of the mean (or the standard deviation of the sampling distribution) in (c) is greater than in (b). As the standard error decreases, values become more concentrated ound the mean. Therefore, the probability that the sample mean will fall close the population mean will always increase when the sample size increases.
The amount of water in a bottle is approximately normally distributed with a mean of 2.50 liters with a standard deviation of 0.025 liter. Complete parts (a) through (e) below. a. What is the probability that an individual bottle contains less than 2.48 liters? 0.212 (Round to three decimal places as needed.) b. If a sample of 4 bottles is selected, what is the probability that the sample mean amount contained is less than 2.48 liters? 0.055 (Round to three decimal places as needed.) c. If a sample of 25 bottles is selected, what is the probability that the sample mean amount contained is less than 2.48 liters? 0.000 (Round to three decimal places as needed.) d. Explain the difference in the results of (a) and (c). Part (a) refers to an individual bottle, which can be thought of as a sample with sample size 2.48|. Therefore, the standard error of the mean for an individual bottle is 0 times the standard error of the sample in (c) with sample size 25. This leads to a probability in part (a) that is larger than the probability in part (c). (Type integers or decimals. Do not round.) e. Explain the difference in the results of (b) and (c). The sample size in (c) is greater than the sample size in (b), so the standard error of the mean (or the standard deviation of the sampling distribution) in (c) is greater than in (b). As the standard error decreases, values become more concentrated ound the mean. Therefore, the probability that the sample mean will fall close the population mean will always increase when the sample size increases.
MATLAB: An Introduction with Applications
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Author:Amos Gilat
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Chapter1: Starting With Matlab
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Transcribed Image Text:The amount of water in a bottle is approximately normally distributed with a mean of 2.50 liters with a standard
deviation of 0.025 liter. Complete parts (a) through (e) below.
a. What is the probability that an individual bottle contains less than 2.48 liters?
0.212 (Round to three decimal places as needed.)
b. If a sample of 4 bottles is selected, what is the probability that the sample mean amount contained is less
than 2.48 liters?
0.055 (Round to three decimal places as needed.)
c. If a sample of 25 bottles is selected, what is the probability that the sample mean amount contained is less
than 2.48 liters?
0.000 (Round to three decimal places as needed.)
d. Explain the difference in the results of (a) and (c).
Part (a) refers to an individual bottle, which can be thought of as a sample with sample size 2.48. Therefore,
the standard error of the mean for an individual bottle is 0 times the standard error of the sample in (c) with
sample size 25. This leads to a probability in part (a) that is larger than
the probability in part (c).
(Type integers or decimals. Do not round.)
e. Explain the difference in the results of (b) and (c).
The sample size in (c) is greater than the sample size in (b), so the standard error of the mean (or the standard
deviation of the sampling distribution) in (c) is greater than in (b). As the standard error decreases, values
become more concentrated around the mean. Therefore, the probability that the sample mean will fall close to
the population mean will always increase when the sample size increases.
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