Example constructing and interpreting a confidence interval for p | AP Statistics | Khan Academy

Khan Academy
18 Dec 201706:35
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TLDRIn this instructional video, the process of estimating the proportion of songs by female artists in a collection is explored. Della, with over 500 songs on her phone, takes a simple random sample of 50 and finds 20 are by female artists. The video guides viewers through the steps to construct a 99% confidence interval, ensuring the sample is random, the sampling distribution is normal, and the independence rule is met. It demonstrates the calculation using the sample proportion, the critical z-value for a 99% confidence level, and the standard error of the sample proportion, ultimately emphasizing the reliability of the interval in reflecting the true population parameter.

Takeaways
  • 🎼 Della has over 500 songs on her mobile phone and wants to estimate the proportion of songs by female artists.
  • 🔬 She takes a simple random sample (SRS) of 50 songs to estimate the proportion.
  • 🎯 20 out of the 50 sampled songs are by female artists, leading to a sample proportion of 0.4.
  • 📊 The goal is to construct a 99% confidence interval for the true population proportion.
  • 🔄 The sampling must be random, which is confirmed by the SRS method used.
  • 📉 The sampling distribution of the sample proportions should be approximately normal, which is checked by ensuring at least 10 successes and 10 failures.
  • 🔄 The independence of observations is assumed by the 10% rule, which is met as the sample size is 10% of the population.
  • 📚 The confidence interval is calculated using the sample proportion plus or minus a critical value times the standard error of the sample proportion.
  • 📐 The standard error is calculated as the sample proportion (p hat) times (1 - p hat) divided by the sample size (n).
  • 📈 The critical value for a 99% confidence level is found using a z-table, looking for the value that leaves 0.5% in each tail.
  • 🔢 The correct critical value is approximately 2.57-2.58, which corresponds to the 99.5th percentile on the z-table.
Q & A

  • What is the purpose of Della's sampling?

    -Della's purpose of sampling is to estimate the proportion of songs on her mobile phone that are by a female artist, as she has over 500 songs and doesn't have time to go through all of them to determine the true population proportion.

  • What does SRS stand for in the context of the script?

    -SRS stands for Simple Random Sample, which is the method Della uses to select 50 songs from her phone for her estimation.

  • How many songs did Della find in her sample that were by a female artist?

    -Della found that 20 out of the 50 songs in her sample were by a female artist.

  • What is the sample proportion denoted as in the script?

    -The sample proportion is denoted as p hat (p̂) in the script.

  • What are the conditions that need to be met before constructing a confidence interval for the sample proportion?

    -The conditions include ensuring the sample is truly random, the sampling distribution of the sample proportions is approximately normal, and the sample size is less than 10% of the population for rough independence.

  • Why is it important to check the normality assumption for the sampling distribution of the sample proportions?

    -Checking the normality assumption is important because it ensures that the confidence interval calculation is valid. It requires at least 10 successes and 10 failures in the sample to meet this condition.

  • What is the independence test or rule mentioned in the script?

    -The independence test or rule, also known as the 10% rule, checks if the sample size is no more than 10% of the population to assume rough independence for each observation, which is important when sampling without replacement.

  • What is the formula for the standard error of the sample proportion?

    -The standard error of the sample proportion is calculated as p̂ * (1 - p̂) / n, where p̂ is the sample proportion and n is the sample size.

  • What does the critical value represent in the context of a confidence interval?

    -The critical value represents the number of standard errors above and below the sample proportion that defines the confidence interval. It is determined by the desired confidence level.

  • Why is the z-table used to find the critical value for a 99% confidence interval?

    -The z-table is used to find the critical value because it provides the z-scores that correspond to specific areas under the standard normal distribution curve, which is needed to calculate the confidence interval for a given confidence level.

  • What does a 99% confidence interval mean in terms of repeated sampling?

    -A 99% confidence interval means that if the sampling and interval construction process were repeated many times, approximately 99% of the intervals would contain the true population parameter.

Outlines
00:00
🎶 Estimating Song Proportions with Confidence Intervals

In this paragraph, the instructor discusses how to estimate the proportion of songs by female artists in Della's mobile phone library of over 500 songs. Della takes a simple random sample (SRS) of 50 songs and finds that 20 are by female artists, leading to a sample proportion (p hat) of 0.4. The instructor explains the process of constructing a 99% confidence interval for the true population proportion (p), emphasizing the importance of checking assumptions such as random sampling, normality of the sampling distribution, and the independence of observations. The normality condition is met with more than 10 successes and failures, and the independence condition is assumed to be met since the sample size is 10% of the population. The formula for the confidence interval is introduced, which includes the sample proportion, a critical value (z star), and the standard error of the sample proportion.

05:02
📊 Determining the Critical Value for a 99% Confidence Interval

The second paragraph focuses on determining the critical value for a 99% confidence interval using a z-table. The instructor clarifies that many z-tables, including those used in exams like AP Stats, include the area up to a certain value, which is why looking up 99.5% is necessary to find the correct z value that corresponds to leaving 0.5% in each tail of the distribution. The instructor guides the viewer to find the z value between 2.57 and 2.58, which is approximately 2.576. This value is then used to calculate the margin of error for the confidence interval. The paragraph concludes by explaining the meaning of a 99% confidence interval in the context of repeated sampling and interval construction, emphasizing that approximately 99% of such intervals would contain the true population parameter.

Mindmap
Keywords
💡SRS (Simple Random Sample)
Simple Random Sample (SRS) refers to a method of selecting a subset of individuals from a larger population such that each individual has an equal chance of being selected. In the video, Della uses an SRS of 50 songs from her collection of 500 to estimate the proportion of songs by female artists. This method is fundamental to the video's theme of statistical sampling and confidence interval estimation.
💡Population Proportion (p)
The population proportion, denoted as 'p', is the true proportion of a particular characteristic in the entire population. In the context of the video, 'p' would represent the actual proportion of songs by female artists in Della's entire collection of 500 songs. The video's narrative revolves around estimating this unknown value through sampling.
💡Sample Proportion (p hat)
The sample proportion, symbolized by 'p hat', is an estimate of the population proportion based on a sample. In the script, Della calculates 'p hat' as 0.4 after finding that 20 out of 50 sampled songs are by female artists. This concept is central to the video's discussion on how to use sample data to infer about the entire population.
💡Confidence Interval
A confidence interval is a range that is likely to contain the true population parameter with a certain level of confidence. The video's main theme involves constructing a 99% confidence interval for the proportion of songs by female artists. The interval is calculated based on the sample proportion and a critical value from a z-table.
💡Normal Distribution
A normal distribution, also known as a Gaussian distribution, is a probability distribution that is symmetric about its mean. The video discusses the assumption that the sampling distribution of the sample proportions is approximately normal, which is a key condition for constructing a valid confidence interval.
💡Sampling Distribution
The sampling distribution is the probability distribution of a given statistic based on a random sample. In the video, the sampling distribution of the sample proportions is discussed as being approximately normal, which allows the use of z-scores for constructing the confidence interval.
💡Standard Error
The standard error is the standard deviation of the sampling distribution of a statistic. In the context of the video, the standard error of the sample proportion is calculated using the formula p hat * (1 - p hat) / n, which is crucial for determining the width of the confidence interval.
💡Critical Value
A critical value is a value from a statistical distribution that defines the threshold for a confidence interval. The video explains that for a 99% confidence interval, the critical value is found using a z-table and corresponds to the point that leaves 0.5% in the tail of the distribution.
💡Z-Table
A z-table is a statistical table that provides the areas under the standard normal curve to the left of a z-score. In the video, the z-table is used to find the critical value needed for constructing the 99% confidence interval for the sample proportion.
💡Independence Test
The independence test, also mentioned as the 10% rule in the video, is a condition that checks whether the sample is independent of the population size. The video explains that if the sample size is no more than 10% of the population, as is the case with Della's sample of 50 out of 500 songs, independence can be assumed.
💡Successes and Failures
In the context of the video, 'successes' refers to the number of songs by female artists in the sample, and 'failures' refers to the number of songs not by female artists. The video mentions that having at least 10 successes and 10 failures is a condition for the sampling distribution to be approximately normal, which Della's sample meets with 20 successes and 30 failures.
Highlights

Della has over 500 songs on her mobile phone and wants to estimate the proportion of songs by a female artist.

She takes a simple random sample (SRS) of 50 songs to estimate the proportion.

20 out of the 50 sampled songs are by a female artist, leading to a sample proportion of 0.4.

The goal is to construct a 99% confidence interval for the proportion of female artist songs.

Assumptions for the sampling distribution to be approximately normal are checked and met.

The sample size meets the condition of having at least 10 successes and 10 failures.

The independence test is considered, assuming no sampling with replacement.

The sample size of 50 is less than 10% of the population, satisfying the independence rule.

The confidence interval will be calculated using the sample proportion plus or minus a critical value.

The critical value is determined by the desired confidence level and the standard error of the sample proportion.

The standard error is calculated using the formula p-hat * (1 - p-hat) / n.

A z-table is used to find the critical value for a 99% confidence level.

The z value for 99.5% is approximately 2.57 to 2.58, indicating the critical value.

The confidence interval is constructed as 0.4 plus or minus 2.576 times the standard error.

The constructed confidence interval will contain the true population parameter in approximately 99% of the cases if the process is repeated.

Transcripts

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Confidence IntervalStatistical TutorialSampling DistributionSRS MethodFemale ArtistsRandom SampleProportion EstimationZ-TableNormal DistributionStatistical AnalysisData Interpretation