Question

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

Answer 1

Answer:

a) $2259.92-2.201\frac{35.569}{\sqrt{12}}=2237.32$    

$2259.92+2.201\frac{35.569}{\sqrt{12}}=2282.52$    

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".

$\bar X$ represent the sample mean for the sample  

$\mu$ 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:

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$   (1)

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

$\bar X= \sum_{i=1}^n \frac{x_i}{n}$ (2)  

$s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}$ (3)  

The mean calculated for this case is $\bar X=2259.92$

The sample deviation calculated $s=35.569$

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:

$df=n-1=12-1=11$

Since the Confidence is 0.95 or 95%, the value of $\alpha=0.05$ and $\alpha/2 =0.025$, 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 $t_{\alpha/2}=2.201$

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

$2259.92-2.201\frac{35.569}{\sqrt{12}}=2237.32$    

$2259.92+2.201\frac{35.569}{\sqrt{12}}=2282.52$    

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