The mean weight of the contents of samples of 30 bags of sugar has
standard error 0.008kg. Choose the option that is closest to the
probability, to three decimal places, that the mean weight of the
contents of samples of 30 bags of sugar will be 1kg or more.
Options for Question
A 0.700 B 0.800 C 0.824
D 0.858 E 0.887 F 0.932
Thanks.
To find the probability that the mean weight of the contents of samples of 30 bags of sugar will be 1kg or more, we need to use the concept of the standard error and the normal distribution.
The standard error represents the variability of the sample mean. In this case, the standard error is given as 0.008kg.
To proceed, we need to assume that the weights of the contents of the bags are normally distributed. With a sample size of 30 bags, we can use the central limit theorem, which states that the distribution of the sample mean approaches a normal distribution as the sample size increases.
Now, we want to find the probability that the mean weight is 1kg or more. We can convert this to a standard normal distribution by subtracting the true mean of the population (unknown in this case) from 1kg, and then dividing by the standard error:
Z = (1kg - true mean) / standard error
Since we don't know the true mean, we will use the fact that the mean of the sample mean is equal to the mean of the population. Therefore, the mean of the sample mean is also 1kg.
Z = (1kg - 1kg) / 0.008kg
Z = 0 / 0.008kg
Z = 0
The probability that Z is less than or equal to 0 is 0.5 since it is the mean of the standard normal distribution. However, we want the probability that the mean weight is 1kg or more, so we need to calculate the complement of this probability:
P(Z ≤ 0) = 0.5
P(Z > 0) = 1 - P(Z ≤ 0) = 1 - 0.5 = 0.5
Therefore, the probability that the mean weight of the contents of samples of 30 bags of sugar will be 1kg or more is closest to 0.500. However, none of the given options match this value exactly.