Chips Ahoy In 1998, as an advertising campaign, the Nabisco...

Chips Ahoy In 1998, as an advertising campaign, the Nabisco Company announced a “1000 Chips Challenge,” claiming that every 18-ounce bag of their Chips Ahoy cookies contained at least 1000 chocolate chips. Dedicated Statistics students at the Air Force Academy (no kidding) purchased some randomly selected bags of cookies, and counted the chocolate chips. Some of their data are given below. (Chance, 12, no. 1[1999]) $$ \begin{array}{cccccccc}{1219} & {1214} & {1087} & {1200} & {1419} & {1121} & {1325} & {1345} \\ {1244} & {1258} & {1356} & {1132} & {1191} & {1270} & {1295} & {1135}\end{array} $$ a) Check the assumptions and conditions for inference. Comment on any concerns you have. b) Create a 95% confidence interval for the average number of chips in bags of Chips Ahoy cookies. c) What does this evidence say about Nabisco’s claim? Use your confidence interval to test an appropriate hypothesis and state your conclusion.

Chips Ahoy In 1998, as an advertising campaign, the Nabisco Company announced a “1000 Chips Challenge,” claiming that every 18-ounce bag of their Chips Ahoy cookies contained at least 1000 chocolate chips. Dedicated Statistics students at the Air Force Academy (no kidding) purchased some randomly selected bags of cookies, and counted the chocolate chips. Some of their data are given below. (Chance, 12, no. 1[1999])
$$
\begin{array}{cccccccc}{1219} & {1214} & {1087} & {1200} & {1419} & {1121} & {1325} & {1345} \\ {1244} & {1258} & {1356} & {1132} & {1191} & {1270} & {1295} & {1135}\end{array}
$$
a) Check the assumptions and conditions for inference. Comment on any concerns you have.
b) Create a 95% confidence interval for the average number of chips in bags of Chips Ahoy cookies.
c) What does this evidence say about Nabisco’s claim? Use your confidence interval to test an appropriate hypothesis and state your conclusion.

Key Concepts

Assumptions for Statistical Inference

When performing inference on parameters such as the population mean, it is crucial to check assumptions such as independence of observations, randomness of sampling, and approximate normality of the data (especially when the sample size is small). This involves ensuring that the data represents a random sample from the population and that any potential outliers or non-normal patterns do not unduly influence the results, which is particularly important when the sample size is less than about 30.

Confidence Interval

A confidence interval provides a range of plausible values for a population parameter based on sample data. In this context, a 95% confidence interval for the average number of chips in cookie bags would give an interval that, if the sampling process were repeated many times, is expected to contain the true mean 95% of the time. It is constructed using the sample mean, standard error, and the appropriate quantile from the t-distribution when the population standard deviation is unknown.

Hypothesis Testing

Hypothesis testing is used to assess claims about a population parameter by comparing sample data to a specified hypothesis. In this scenario, one tests the claim made by Nabisco by setting up a null hypothesis that the true mean number of chips is at least 1000 against an alternative hypothesis that it is lower or different than expected. The confidence interval is often used in this context to determine if the claimed parameter value falls within the range of plausible values based on the data.

t-Distribution and t-Procedure

When the sample size is relatively small and the population standard deviation is unknown, the t-distribution is used instead of the normal distribution to account for additional uncertainty. The t-procedure involves using a t-distribution with n-1 degrees of freedom to construct confidence intervals and perform hypothesis tests for the population mean. This approach helps to ensure that the inference is appropriately calibrated given the sample characteristics.

Transcript

00:01 This is the question that we have this time.

00:04 Nabisco company claims that every bag, their every 18 ounce bag of ahoy chip cookies, contains at least 1 ,000 chips.

00:13 All right.

00:14 And this is the sample that is given to us.

00:16 Sample size is 16 and we are given these readings.

00:20 The first one says check the assumptions.

00:21 Let us look at a histogram.

00:24 You see that i am going to generate a histogram here.

00:27 And we can see that since the sample size is small, this looks like.

00:32 What we can say a one -sided histogram but were the samples size to increase i'm sure that it would look something like this and this would be pretty normal so we can say that yes it is unimodal we can see that in the middle there are 4 4 4 and then there is 3 there is a 1 and there would also be similar kind of a trend on the left -hand side if we had more number of samples so yes it does look normal so we can say that the conditions for inference and the assumptions are satisfied and plus all of these are randomly selected which is very important.

01:08 The second one is create a 95 % confidence interval.

01:10 What is the formula to create a 95 % confidence interval? it is x bar that is the mean of the sample plus minus t statistic.

01:18 How do we find the t critical value for t in this case? our degree of freedom is 50 and our confidence level is 95...

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