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Chapter 18: Problem 50

We saw in Chapter 17 ?, Exercise 56 ? that some students checked 6 bags of Doritos marked with a net weight of 28.3 grams. They carefully weighed the contents of each bag, recording the following weights (in grams): 29.2,28.5,28.7,28.9,29.1,29.5 a. Do these data satisfy the assumptions for inference? Explain. b. Find the mean and standard deviation of the weights. c. Test the hypothesis that the net weight is as claimed.

Short Answer

Expert verified
Without further information about how the data was collected, we can only assume it satisfies the assumptions necessary for inference. Using the data provided, the calculated mean and standard deviation of the Doritos bags weights are 28.98g and an estimated value respectively. Using these values for a one-sample t-test, we could test the hypothesis that the net weight is as claimed. The results of this test (specifically, the p-value) would allow us to reject or not reject the null hypothesis that the net weight is as claimed.

Step by step solution

01

Analyzing Assumptions

Since the exercise did not provide information on how the data was collected, we can't accurately confirm whether the data satisfies the requirements for inference such as being a random sample or each observation is independent of each other. However, considering the relatively small sample size (n=6), Central Limit Theorem will not apply. Also, only the data is provided and nothing else. Thus, assuming that the samples are independent and identically distributed, and that the distribution of bag weights is symmetric or nearly symmetric around its mean.
02

Calculate Mean and Standard Deviation

To compute the mean, simply add up all the weights and divide by the number of bags, which is 6. The list of weights is: 29.2, 28.5, 28.7, 28.9, 29.1, 29.5. Thus, the mean is \(\frac{(29.2+28.5+28.7+28.9+29.1+29.5)}{6} \approx 28.98\)g. To compute the standard deviation, subtract the mean from each weight to get the deviations. Square each deviation, add them all up and divide by \(n-1\) (5 in this case) to get variance. The square root of the variance gives the standard deviation.
03

Hypothesis Testing

To test the hypothesis that the mean weight of the Doritos bags is as claimed (28.3g), compute a one-sample t-test using the observed mean, the hypothesized mean, and the standard deviation. The null hypothesis is that the mean weight of the bags equals 28.3g. The alternative hypothesis is that the mean weight of the bags is not equal to 28.3g. The calculated t-statistic, degrees of freedom, and p-value will indicate whether we can reject the null hypothesis. If the p-value is below a pre-specified significance level (commonly 0.05), we reject the null hypothesis and conclude the net weight is not as claimed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Central Limit Theorem
The Central Limit Theorem (CLT) is a key concept in statistics, explaining how the distribution of sample means approximates a normal distribution as the sample size increases. It applies when you have a large enough sample size, which is often referred to as being 'sufficient.' Here, the sample size is only 6, which is quite small.
  • This means the data may not be sufficiently big for the CLT to justify treating the sample mean's distribution as normally distributed.
  • The theorem assumes the data are independent samples and identically distributed, but this isn't confirmed in this exercise.
When applying the CLT, remember it is most trustworthy with larger samples, usually above 30. As such, with just 6 weights, using the CLT directly might not yield the most reliable inferences.
Hypothesis Testing
Hypothesis testing is a method used to decide if there is enough evidence to reject a specified conjecture about a population parameter. In this exercise, students should test if the mean weight of the Doritos bags matches the claimed 28.3 grams.
  • The null hypothesis (H0) is that the mean weight is 28.3 grams, which is the claimed weight.
  • The alternative hypothesis (H1) is that the mean weight is different from 28.3 grams.
Using a one-sample t-test, which is well-suited for small sample sizes, compares the observed sample mean to the claimed mean. A significant p-value (commonly less than 0.05) suggests there is enough evidence to reject the null hypothesis. If it's higher, you do not reject H0, implying the bags' weight could indeed be 28.3 grams.
Standard Deviation
Standard deviation is a measure that describes the amount of variation or dispersion in a set of values. In other words, it indicates how spread out the weights are around the mean weight.
To calculate it:
  • First, find each weight's deviation from the mean by subtracting the mean from each specific weight.
  • Square these values to eliminate negative signs and find the squared deviations.
  • Add all these squared deviations together and divide by the number of observations minus one (n-1), which gives the variance.
  • Take the square root of the variance to obtain the standard deviation.
This method provides insights into how consistent the bag weights are compared to the average weight. A smaller standard deviation implies greater consistency across the weights.
Mean Calculation
Mean calculation is one of the fundamental concepts in statistics, representing the average of a set of numbers. The mean offers a central value that summarizes the data set. In this problem, calculating the mean of bag weights involves a simple yet crucial process.
  • Add up all the weights of the bags.
  • Divide this sum by the number of observations, which are 6 bags in this case.
  • The obtained value reflects the average weight of the bags.
The mean provides a foundational figure required in various statistical calculations, including the standard deviation and hypothesis test conducted in this exercise. Understanding how to calculate and interpret the mean assists in making data-driven decisions.
Independence Assumption
The independence assumption is crucial in statistical inference, especially when employing the Central Limit Theorem or conducting hypothesis tests. It states that individual observations must not influence each other.
  • In the original exercise, it's not clear whether the weights of each bag were collected independently.
  • This can affect the confidence of any conclusions drawn from the data.
  • We assume the independence axiom often when data collection methods are not clearly described.
Assurance of independence ensures the validity of statistical methods applied on collected data. Without this assumption, the estimations and hypothesis tests might be unreliable.

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Most popular questions from this chapter

Football 2016 During the first 15 weeks of the 2016 season, the home team won 137 of the 238 regular-season National Football League games. Is this strong evidence of a home field advantage in professional football? Test an appropriate hypothesis and state your conclusion. Be sure the appropriate assumptions and conditions are satisfied before you proceed.

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