The weights of four randomly and independently selected bags of potatoes labeled 25 pounds were found to be 25.9, 25.4, 26.0, and 26.2
pounds. Assume Normality. Answer parts (a) and (b) below.
a. Find a 95% confidence interval for the mean weight of all bags of potatoes.
First, notice that the population standard deviation is unknown. This means that you can use a one-sample t-interval to find the confidence
interval.
The formula below can be used to find the confidence interval where x is the sample mean, the multiplier t* is a constant found using
a t-distribution with n - 1 degrees of freedom that is used to fine-tune the margin of error so it has the desired level of confidence, s is the
sample standard deviation, and n is the sample size.
$$x \pm t^* \frac{s}{\sqrt{n}}$$
While the above formula or technology can be used to compute the confidence interval, for this example, use technology.
Enter the four given values into your technology. Use technology to calculate a 95% confidence interval.
The lower bound of the confidence interval is 25.33, rounded to the nearest hundredth.
The upper bound of the confidence interval is 26.42, rounded to the nearest hundredth.
This means that we are 95% confident the population mean weight is between 25.33 and 26.42 pounds.
D. Can you reject a population mean of 25 pounds? Explain.
In order for a confidence interval to be valid, the following three conditions must hold.
Condition 1: Random Sample and Independence. The data must be collected randomly, and
each observation must be independent of the others.
Condition 2: Large sample. Either the population must be Normally distributed or the sample
size must be at least 25.
Condition 3: Big Population. If the sample is collected without replacement, then the
population size must be at least 10 times larger than the sample size.
tice that the three conditions are met since the sample was randomly drawn from an independent data set, the population is
rmally distributed, and the population of all bags of potatoes is at least 10 times larger than the sample size of 4. Also notice that the value
is outside of and is not captured by the 95% confidence interval (25.33,26.42).
ce the conditions are met, the confidence interval (25.33,26.42) can be accurately used. Remember that the purpose of a confidence
erval is to give a range of values within which the mean of the entire population is likely to lie. Use this to determine if a possible population
an of 25 pounds can be rejected.