Question

The amount of water in a bottle is approximately normally distributed with a mean of 2.85 litres with a standard deviation of 0.035-liter.

b. If a sample of 4 bottles is​ selected, the probability that the sample mean amount contained is less than 2.82 ​litres is 0.043.

c. If a sample of 25 bottles is​ selected, the probability that the sample mean amount contained is less than 2.82 ​litres is 0.

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 (Pick either in [More than] or [Less than]) in​ (b).

As the standard error (Pick either in [Increase] or [Does not change] or [Decreases]), values become more concentrated around the (Pick either in [standard deviation] or [median] or [mean]) .​ Therefore, the probability that the sample mean will fall (Pick either in [Far] or [Close]) to the population mean will always increase when the sample size (Pick either in [increases] or [decreases])

Answer 1

The variable of interest is amount of water in the bottle. Let X represents the amount of water in the bottle which is normally distributed with mean of 2.85 litres and standard deviation of 0.035 litres. Then the probability density function is given by

Let us consider the property associated to standard normal distribution Z as

1. P(Z>0)=P(Z<0)=0.5

2. P(Z<-a)=P(Z>a)=0.5-P(0<Z<a)

1. Let us consider that a sample of 4 bottle is selected, then we have to calculate the probability of the sample mean amount contained less than 2.82 litres, which is given below

Therefore the probability is 0.0432.

2. If a sample of 25 bottle is selected then the probability that the sample mean amount is less than 2.82 is given by

Therefore the probability is 0 that the sample mean of 25 students is less than 2.82.

The probability in normal distribution is generally obtained by the points or the region faaling in the confidence interval. As the sample size increases the standard deviation decreases and hence the confidence interval decreases as which can be seen from the graph

In the above red one indicates normal distribution with lower standard deviation than black one.

   Thus it can be summarise as

The sample size in​ (c) is greater than the sample size in​ (b), so the standrad deviation in​ (c) which is 0.007 is less than then the stndard deviation  in​ (b) which is 0.0175. Therefore, it concluded that as the standard error or simply standard deviation decreases,the obtained desired values become more concentrated around the mean .

In case of Normal distribution mean= median = mode as all the three central tendencies coincide so one can say it is concentrated around mean or median.

​ Therefore, the probability that the sample mean will fall close to the population mean will always increase when the sample size increases.