The compressive strength of concrete is being tested by a civil engineer who tests 12 specimens and obtains the following data: 8) 2216 2237 2225 2301 2318 2255 9) 2249 2204 2281 2263 2275 2295 a) Construct a 95% two-sided confidence interval on the mean strength. b) What assumptions, if any, are needed to construct the above interval. c) Interpret the interval

a)    

So on this case the 95% confidence interval would be given by (2237.32;2282.52)    

b) 1) Randomization we assume that the sample is selected from a random sample

2) Independence we assume that each observation is independent from the other ones

3) Sample size condition the sampe needs to be less than 10% of the population size.

c) For this case we can conclude with 95% of confidence that the true mean of compressive strength of concrete is between 2237.32 and 2282.52

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

represent the sample mean for the sample  

population mean (variable of interest)

s represent the sample standard deviation

n represent the sample size  

Part a

The confidence interval for the mean is given by the following formula:

  (1)

In order to calculate the mean and the sample deviation we can use the following formulas:  

(2)  

(3)  

The mean calculated for this case is

The sample deviation calculated

In order to calculate the critical value we need to find first the degrees of freedom, given by:

Since the Confidence is 0.95 or 95%, the value of and , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,11)".And we see that

Now we have everything in order to replace into formula (1):

So on this case the 95% confidence interval would be given by (2237.32;2282.52)    

Part b

1) Randomization we assume that the sample is selected from a random sample

2) Independence we assume that each observation is independent from the other ones

3) Sample size condition the sampe needs to be less than 10% of the population size.

Part c

For this case we can conclude with 95% of confidence that the true mean of compressive strength of concrete is between 2237.32 and 2282.52