Example constructing a t interval for a mean | Confidence intervals | AP Statistics | Khan Academy | Summary and Q&A

TL;DR

A nutritionist uses a sample of 14 burritos to construct a 95% confidence interval for the mean caloric content, with a sample mean of 700 calories and a sample standard deviation of 50 calories.

Key Insights

• 🌯 The nutritionist wants to estimate the mean caloric content of burritos at a popular restaurant.
• 😒 They use a sample size of 14 burritos and calculate a sample mean of 700 calories with a sample standard deviation of 50 calories.
• 👷 A T statistic is used to construct the 95% confidence interval.
• ❓ The margin of error for the confidence interval is approximately 28.9 calories.
• ❓ The conditions for a valid confidence interval are met, including random sampling, rough normality of the sampling distribution, and independence.
• 🧡 The confidence interval is a range within which the true mean caloric content is likely to fall.
• 👻 Constructing a confidence interval provides a level of uncertainty and allows for inferences about the population mean.

Transcript

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Questions & Answers

Q: How does the nutritionist estimate the mean caloric content of burritos at a popular restaurant?

The nutritionist takes a random sample of 14 burritos and measures their caloric content. They calculate the sample mean (700) and sample standard deviation (50), then use these values along with a T statistic to construct a 95% confidence interval.

Q: What is the purpose of constructing a confidence interval in this scenario?

The purpose is to estimate the range within which the true mean caloric content of the burritos at the restaurant is likely to fall. This provides a level of uncertainty and allows for generalizations to the population.

Q: Why does the nutritionist use a T statistic instead of a Z statistic?

A T statistic is used because the nutritionist does not know the actual standard deviation of the population. If the population standard deviation was known, a Z statistic would be used instead.

Q: What are the conditions for a valid confidence interval in this context?

The conditions include taking a random sample, having a roughly normal sampling distribution (with no significant outliers), and meeting the independence condition (assuming sampling without replacement and the sample size being less than 10% of the population).

Summary & Key Takeaways

• A nutritionist wants to estimate the average caloric content of burritos at a popular restaurant.

• They take a random sample of 14 burritos and measure their caloric content.

• Using the sample mean, sample standard deviation, and a T statistic, they construct a 95% confidence interval for the mean caloric content.