A random sample of 120 apricots is selected from trees in a Hawkes Bay orchard and the weight of each apricot is measured, giving mean of 42 grams and SD of 12 grams.This information is to be used to calculate a 95% confidence interval for the true mean weight of apricots grown on trees in this Hawkes Bay orchard.Use the drop-down menus to identify if the following conditions for constructing this confidence interval are satisfied or not.The sample is representative of the population.AnswerThe sampling distribution of the sample mean is normal.
Question
A random sample of 120 apricots is selected from trees in a Hawkes Bay orchard and the weight of each apricot is measured, giving mean of 42 grams and SD of 12 grams.This information is to be used to calculate a 95% confidence interval for the true mean weight of apricots grown on trees in this Hawkes Bay orchard.Use the drop-down menus to identify if the following conditions for constructing this confidence interval are satisfied or not.The sample is representative of the population.AnswerThe sampling distribution of the sample mean is normal.
Solution
- The sample is representative of the population.
This condition is generally assumed to be satisfied in statistical analysis when a random sample is taken. In this case, the problem states that a random sample of 120 apricots is selected from trees in a Hawkes Bay orchard. Therefore, we can assume that this condition is satisfied.
Answer: Satisfied
- The sampling distribution of the sample mean is normal.
The Central Limit Theorem (CLT) states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large (usually n > 30).
In this case, the sample size is 120, which is greater than 30. Therefore, according to the CLT, the sampling distribution of the sample mean is approximately normal.
Answer: Normal
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