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Chapter 20: Problem 65

The data below show the sugar content (as a percentage of weight) of several national brands of children鈥檚 and adults鈥� cereals. Create and interpret a 95% confidence interval for the difference in mean sugar content. Be sure to check the necessary assumptions and conditions Children's cereals: 40.3, 55, 45.7, 43.3, 50.3, 45.9, 53.5, 43,44.2,44,47.4,44,33.6,55.1,48.8,50.4,37.8,60.3,46.6 Adults' cereals: \(20,30.2,2.2,7.5,4.4,22.2,16.6,14.5,\) \(21.4,3.3,6.6,7.8,10.6,16.2,14.5,4.1,15.8,4.1,2.4,3.5,\) 8.5,10,1,4.4,1.3,8.1,4.7,18.4

Short Answer

Expert verified
The solution requires calculation of means, standard deviations and finally the 95% confidence interval for both groups of data. The interpretation of the confidence interval will form the final part of the answer.

Step by step solution

01

Calculate Means

Firstly, calculate the mean of the sugar content for both, the children's and adults' cereal. The formula to use is: Mean = Sum of data points / Total number of data points.
02

Calculate the Standard Deviation

Next, the standard deviation should be calculated for both children's and adults' cereals. Use all data points, the calculated mean and the corresponding formula: Standard Deviation = sqrt[ 危 ( xi - mean )虏 / (n)].
03

Analyze and Verify Conditions

Verify the necessary assumptions and conditions. The necessary conditions for applying the confidence interval are: 1. Both of the samples are independent of each other. 2. Both samples are approximately normally distributed, or the sample sizes are big enough.
04

Calculate the Confidence interval

Calculate the confidence interval for the difference. The formula for a Confidence Interval is: CI = mean1 - mean2 卤 Z * sqrt[ (sd1虏/n1) + (sd2虏/n2) ]. Here, Z value is determined by the desired level of confidence. For 95%, Z=1.96.
05

Interpret the Results

Finally, present and interpret the results of the calculated confidence interval. The interpretation will show if there is a significant difference in the sugar content between children's cereals and adults' cereals.

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Key Concepts

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

Mean Sugar Content
Understanding the mean sugar content is essential for nutrition analysis and comparing different food products, such as children's and adults' cereals. The mean, or average, represents the central value of a data set and is calculated by adding up all the individual sugar content percentages and then dividing by the total number of cereals measured.

For instance, if we have a set of sugar content values for children's cereals, we add up all the percentages and divide by the number of cereals to find the average sugar content. This single number gives us a quick overview of the overall sweetness level in the cereals aimed at children compared to those targeted at adults, which are typically perceived to have a lower sugar content.
Standard Deviation
The standard deviation is a measure of how spread out the values in a data set are around the mean. In simpler terms, it tells us how much variation or dispersion there is from the average (mean) sugar content in our cereal samples.

To calculate the standard deviation, each sugar content value is compared to the mean by finding the difference, squaring that difference, and then averaging those squared differences over the number of data points. The square root of this average gives us the standard deviation. A lower standard deviation indicates the sugar content of the cereals is consistently close to the mean, while a higher standard deviation suggests a wider range of sugar levels.

Knowing the standard deviation can help identify outliers in cereal sweetness and assess if certain brands vary significantly from the average.
Statistical Significance
Statistical significance is a determination of whether the observed difference in parameters, such as mean sugar content between two groups, is due to chance or a specific cause. When we say a result is 'statistically significant,' we mean that it's unlikely that this result occurred by random chance alone and thus reflects a true difference between the groups being compared.

After calculating the 95% confidence interval for the difference between the mean sugar contents of children's and adults' cereals, we look to see if this interval includes zero. If zero is not within the interval, this suggests that the difference in means is significant 鈥� meaning that children's and adults' cereals likely have a true difference in their mean sugar contents.

Considering our exercise, we check for statistical significance to validate whether the apparent disparity in sweetness between the two cereal types is reliable and not just a fluctuation due to sample variability.

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Most popular questions from this chapter

A man who moves to a new city sees that there are two routes he could take to work. A neighbor who has lived there a long time tells him Route A will average 5 minutes faster than Route B. The man decides to experiment. Each day, he flips a coin to determine which way to go, driving each route 20 days. He finds that Route A takes an average of 40 minutes, with standard deviation 3 minutes, and Route B takes an average of 43 minutes, with standard deviation 2 minutes. Histograms of travel times for the routes are roughly symmetric and show no outliers. a. Find a \(95 \%\) confidence interval for the difference in average commuting time for the two routes. (From technology, \(d f=33.1 .)\) b. Should the man believe the old-timer's claim that he can save an average of 5 minutes a day by always driving Route A? Explain.

Egyptians Some archaeologists theorize that ancient Egyptians interbred with several different immigrant populations over thousands of years. To see if there is any indication of changes in body structure that might have resulted, they measured 30 skulls of male Egyptians dated from 4000 B.c.E. and 30 others dated from 200 B.C.E. (A. Thomson and R. Randall-Maciver, Ancient Races of the Thebaid, Oxford: Oxford University Press, 1905\()\) a) Are these data appropriate for inference? Explain. b) Create a \(95 \%\) confidence interval for the difference in mean skull breadth between these two eras. c) Do these data provide evidence that the mean breadth of males' skulls changed over this period? Explain. d) Perform Tukey's test for the difference. Do your conclusions of part c change? \({ }^{*}\) e) Perform a rank sum test for the difference. Do your conclusions of part c change?

A study published in the Archives of General Psychiatry examined the impact of depression on a patient's ability to survive cardiac disease. Researchers identified 450 people with cardiac disease, evaluated them for depression, and followed the group for 4 years. Of the 361 patients with no depression, 67 died. Of the 89 patients with minor or major depression, 26 died. Among people who suffer from cardiac disease, are depressed patients more likely to die than non-depressed ones? a. What kind of design was used to collect these data? b. Write appropriate hypotheses. c. Are the assumptions and conditions necessary for inference satisfied? d. Test the hypothesis and state your conclusion. e. Explain in this context what your \(\mathrm{P}\) -value means. \(f\). If your conclusion is actually incorrect, which type of error did you commit? g. Create a 95\% Cl. h. Interpret your interval. i. Carefully explain what "95\% confidence" means.

A consumer magazine plans to poll car owners to see if they are happy enough with their vehicles that they would purchase the same model again. They'll randomly select 450 owners of American-made cars and 450 owners of Japanese models. Obviously, the actual opinions of the entire population couldn't be known, but suppose \(76 \%\) of owners of American cars and \(78 \%\) of owners of Japanese cars would purchase another. a. What kind of sampling design is the magazine planning to use? b. What difference would you expect their poll to show? c. Of course, sampling error means the poll won't reflect the difference perfectly. What's the standard deviation for the difference in the proportions? d. Sketch a sampling model for the difference in proportions that might appear in a poll like this. e. Could the magazine be misled by the poll, concluding that owners of American cars are much happier with their vehicles than owners of Japanese cars? Explain.

A Time magazine article about a survey of men's attitudes reported that 11 of 161 black respondents and 20 of 358 Latino respondents responded "Yes" to the question "Are you a stay-at-home dad?" How big is the difference in proportions in the two populations? a. Construct and interpret an appropriate confidence interval. b. Overall, the survey contacted 1302 men and claims a margin of error of \(\pm 2.9 \%\). Why is the margin of error different for your confidence interval?
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