A sample of 100 iron bars is said to be drawn from a large number of bars whose lengths are normally distributed with mean 4 feet and standard deviation 6 ft .If the sample mean is 4.2 feet can the sample be truly regarded as random sample
To determine if the sample can be truly regarded as a random sample, we need to conduct a hypothesis test.
Null hypothesis (H₀): The sample mean is equal to the population mean (µ = 4).
Alternative hypothesis (H₁): The sample mean is not equal to the population mean (µ ≠ 4).
We will use a significance level of 0.05 to make our decision.
First, we calculate the standard error of the mean:
Standard error = (standard deviation)/(√sample size)
Standard error = 6/(√100)
Standard error = 6/10
Standard error = 0.6 feet
Next, we calculate the z-score:
z = (sample mean - population mean)/(standard error)
z = (4.2 - 4)/(0.6)
z = 0.2/0.6
z = 0.333
Using a z-table or a statistics calculator, we find that the probability of obtaining a z-score of 0.333 or less is approximately 0.6293. This is greater than our significance level of 0.05.
Since the probability is greater than our significance level, we fail to reject the null hypothesis. Therefore, we can conclude that the sample mean of 4.2 feet can be considered as a random sample from the population of iron bars whose lengths are normally distributed with a mean of 4 feet and a standard deviation of 6 feet.