1)Suppose that the weight of tomato juice in mechanically filled cans follows a normal distribution with a mean of 464 grams and a standard deviation of 8 grams.
a)A random sample of size n = 18 cans is taken. Will the sampling distribution of the sample mean weight be normal? Why or why not, or can it not be determined?
The larger the sample, the more likely it will approximate a normal distribution.
To determine whether the sampling distribution of the sample mean weight will be normal, we need to consider the Central Limit Theorem (CLT).
According to the CLT, when the sample size is sufficiently large (usually n ≥ 30), regardless of the shape of the population distribution, the sampling distribution of the sample mean will be approximately normal.
In this case, the sample size is n = 18, which is less than 30. Therefore, we cannot conclude that the sampling distribution of the sample mean weight will be normal based solely on the sample size.
However, if the original population distribution is reasonably close to a normal distribution and there are no extreme outliers or skewness present, the sampling distribution of the sample mean is likely to be approximately normal even when the sample size is smaller than 30.
In summary, given that the weight of tomato juice in mechanically filled cans follows a normal distribution, and assuming no extreme outliers or skewness, we can expect that the sampling distribution of the sample mean weight will be approximately normal, despite the sample size being smaller than 30.