Ahigh school statistics class wants to estimate the average number of chocolate chips in a generic brand of chocolate chip cookies. they collect a random sample of cookies, count the chips in each cookie, and calculate a 95% confidence interval for the average number of chips per cookie (18.6 to 21.3).the next four statements present four different interpretations of these results. indicate if each interpretation is valid or invalid. interpretation #1: we are 95% certain that each cookie of this brand has approximately 18.6 to 21.3 chocolate chips. -- #2: we expect 95% of the cookies to have between 18.6 and 21.3 chocolate chips. -- #3: we would expect about 95% of all possible sample means from this population to be between 18.6 and 21.3 chocolate #4: we are 95% certain that the confidence interval of 18.6 to 21.3 includes the true average number of chocolate chips per cookie. validinvalid

#1: We are 95% certain that each cookie of this brand has approximately 18.6 to 21.3 chocolate chips.

Thats not false at all but present a problem, no mention the parameter of interest on this case the mean for this case is not the most valid conclusion.

#2: We expect 95% of the cookies to have between 18.6 and 21.3 chocolate chips.

Like the option #1 is not FALSE but we see the same problem the statement not present the conclusion in terms of the parameter of interest on this case the mean.

#3: We would expect about 95% of all possible sample means from this population to be between 18.6 and 21.3 chocolate chips.

That's not at all true, since the statement is related to the possible samples and not to the confidence level and the parameter of interest, so for this case is not the best option.

#4: We are 95% certain that the confidence interval of 18.6 to 21.3 includes the true average number of chocolate chips per cookie.

That's the best interpretation since we have the confidence level used to construct the interval, the results are related to the parametr of interest, so it's the most complete answer for this case.

Step-by-step explanation:

1) 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  

2) Solution to the problem

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

  (1)

The 95% confidence interval obtained on this case is (18.6 ; 21.3)

Based on this we can analyze one by one the statements like this:

#1: We are 95% certain that each cookie of this brand has approximately 18.6 to 21.3 chocolate chips.

Thats not false at all but present a problem, no mention the parameter of interest on this case the mean for this case is not the most valid conclusion.

#2: We expect 95% of the cookies to have between 18.6 and 21.3 chocolate chips.

Like the option #1 is not FALSE but we see the same problem the statement not present the conclusion in terms of the parameter of interest on this case the mean.

#3: We would expect about 95% of all possible sample means from this population to be between 18.6 and 21.3 chocolate chips.

That's not at all true, since the statement is related to the possible samples and not to the confidence level and the parameter of interest, so for this case is not the best option.

#4: We are 95% certain that the confidence interval of 18.6 to 21.3 includes the true average number of chocolate chips per cookie.

That's the best interpretation since we have the confidence level used to construct the interval, the results are related to the parametr of interest, so it's the most complete answer for this case.

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true

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