Critical value (z*) for a given confidence level | AP Statistics | Khan Academy

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15 Dec 201706:21

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TLDRThe video script explains the process of constructing a one-sample z interval to estimate the proportion of defective computers in a factory. Elena, the instructor, uses a 94% confidence level with a sample of 200 computers, finding 12 defects. The focus is on determining the critical z value (z star) for the interval. The script clarifies the concept of confidence intervals, the importance of the z value in capturing the desired area under the curve, and guides through the process of finding the z value from a z-table, concluding with a z star of 1.88 for the 94% confidence interval.

Takeaways

📚 Elena is constructing a one-sample z interval to estimate the proportion of computers with a defect at a factory.
🔢 She has chosen a confidence level of 94% for her interval estimation.
💻 A sample of 200 computers revealed that 12 had the defect, which will be used to calculate the sample proportion.
📉 The concept of a confidence interval is explained, emphasizing its purpose to estimate the true population parameter.
🎯 The script explains that a 94% confidence interval means that 94% of such intervals would contain the true population parameter if the sampling were repeated many times.
📊 The critical value, z star, is discussed as the number of standard deviations that define the width of the confidence interval.
🔍 The standard error of the statistic is mentioned as a component in calculating the confidence interval, but it's not the focus of the critical value determination.
📚 The importance of using a z-table or calculator correctly is highlighted, especially considering the type of area (one-tailed or two-tailed) the table or calculator provides.
📈 The process of finding z star involves looking for the z value that leaves 3% in the tails of the distribution, corresponding to the 94% confidence level.
🔢 The script provides a practical example of looking up the z value in a z-table, identifying 1.88 as the critical value for a 94% confidence interval.
📝 The final takeaway is the identification of the critical z value, z star, as 1.88 for constructing Elena's confidence interval.

Q & A

What is the purpose of a one-sample z interval in statistics?
-A one-sample z interval is used to estimate a population parameter, such as the proportion of computers with a certain defect in this case, when the true population parameter is unknown.
What is the significance of the 94% confidence level in this scenario?
-A 94% confidence level means that if we were to repeatedly construct confidence intervals using random samples, approximately 94% of these intervals would contain the true population proportion.
What is the sample size in Elena's study of computer defects?
-Elena's sample size is 200 computers.
How many computers with defects were found in Elena's sample?
-In the sample of 200 computers, 12 were found to have the defect.
What is the critical value z star used for in constructing a confidence interval?
-The critical value z star is used to determine the number of standard deviations above and below the sample statistic to capture the desired confidence level.
Why is it necessary to look up the critical value z star in a z-table or calculator?
-The critical value z star is needed to find the number of standard deviations that correspond to the desired confidence level, which is not known a priori and must be determined from a standard normal distribution.
What is the standard error of the statistic in the context of this script?
-The standard error is the standard deviation of the sampling distribution of the sample proportion, which is used in calculating the confidence interval.
How is the standard error of the sample proportion calculated?
-The standard error is calculated as the square root of the product of the sample proportion and its complement, divided by the sample size.
What does the z-table represent and how is it used?
-A z-table represents the cumulative area under the standard normal curve up to a certain number of standard deviations from the mean. It is used to find the critical value z star that corresponds to the desired confidence level.
How does the symmetry of the normal distribution affect the calculation of the confidence interval?
-The symmetry means that the confidence interval is calculated by taking the sample statistic plus or minus the critical value z star times the standard error, ensuring that the desired confidence level is maintained.
What is the critical value z star that Elena should use for her 94% confidence interval?
-Elena should use a critical value z star of 1.88 for her 94% confidence interval.

Outlines

00:00

📊 Understanding Confidence Intervals and Critical Values

In this paragraph, the instructor discusses the concept of a one-sample z interval used to estimate the proportion of computers with a certain defect at a factory. Elena, the subject of the scenario, selects a 94% confidence level for her interval. The instructor explains the purpose of confidence intervals and how they are used to estimate the true population parameter, in this case, the proportion of defective computers. A random sample of 200 computers reveals 12 defects, leading to the calculation of a sample proportion. The instructor emphasizes the importance of the critical value, z star, in constructing the confidence interval, which represents the number of standard deviations from the mean that capture 94% of the probability. The explanation includes the process of looking up z values on a z-table and the significance of the symmetric distribution of the remaining 6% of the area around the mean.

05:05

🔍 Calculating the Critical Z Value for a 94% Confidence Interval

The instructor continues the discussion by focusing on finding the specific critical z value, z star, needed for a 94% confidence interval. Using a z-table provided for AP Statistics, the instructor demonstrates how to locate the z value that corresponds to 97% of the area under the standard normal curve, which leaves 3% in the tails, thus achieving the desired 94% confidence level. The z-table is used to find the closest z value to 97%, which is approximately 1.88. This z value is then used to construct the confidence interval, indicating that 94% of such intervals would contain the true population parameter if the sampling were repeated multiple times.

Mindmap

Keywords

💡Confidence Interval

A confidence interval is a range of values that is likely to contain the true population parameter. In the video, it is used to estimate the proportion of computers with a defect. The script explains that a 94% confidence interval means that if the process of creating intervals is repeated many times, approximately 94% of those intervals will contain the true population proportion of defective computers.

💡Population Proportion

Population proportion refers to the true value of a characteristic in an entire population. In the context of the video, the population proportion is the actual fraction of computers with a defect that the factory produces. The script emphasizes that while this true value is unknown, it is what the sample proportion is trying to estimate.

💡Sample Proportion

Sample proportion is the ratio of the number of successes (e.g., defective computers) to the total number of observations in a sample. In the script, the sample proportion is calculated from a random sample of 200 computers, where 12 were found to have a defect, and is used to estimate the population proportion.

💡Critical Value

The critical value, often denoted as z*, is a number from the standard normal distribution that defines the bounds of a confidence interval. The script explains that for a 94% confidence interval, the critical value z* is determined by looking up the value that leaves 3% in the tails of the distribution, which corresponds to capturing 94% of the area under the curve.

💡Z-Table

A z-table is a statistical tool used to find the critical values for a given confidence level. The script describes how to use a z-table to find the critical value z* for a 94% confidence interval by looking for the value that corresponds to 97% of the area under the standard normal curve, leaving 3% in the tails.

💡Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the video, the standard deviation of the sampling distribution is used in the formula for the confidence interval. The script mentions multiplying the critical value by the standard error of the statistic, which is related to the standard deviation.

💡Standard Error

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 used to calculate the margin of error in the confidence interval. The script explains that it is an important component in the formula for constructing the interval.

💡Sampling Distribution

A sampling distribution is the probability distribution of a given statistic based on a random sample. The script discusses the sampling distribution of the sample proportion, which is assumed to be roughly normal for the purpose of constructing the confidence interval.

💡True Population Parameter

The true population parameter is the actual value of a characteristic for the entire population. The script uses the term to refer to the unknown true proportion of defective computers that the sample proportion is trying to estimate.

💡Random Sample

A random sample is a subset of a population where each member has an equal chance of being selected. In the video, a random sample of 200 computers is taken to estimate the proportion of computers with a defect. The script emphasizes that this sample is crucial for creating an accurate confidence interval.

💡Margin of Error

Margin of error is the range within which the true value is estimated to lie, given a certain level of confidence. In the script, the margin of error is calculated as the critical value z* times the standard error, and it represents the uncertainty in the estimate of the population proportion.

Highlights

Elena aims to construct a one-sample z interval to estimate the proportion of defective computers in a factory.

Confidence level chosen is 94%, which implies that 94% of such intervals will contain the true population parameter.

A random sample of 200 computers reveals 12 defective units.

The concept of a critical value z star is introduced for constructing the confidence interval.

The critical value z star is determined by the desired confidence level and the standard deviation of the sampling distribution.

The true population proportion is unknown, hence it is estimated through the sample proportion.

The standard error of the statistic is used in the calculation of the confidence interval.

The sampling distribution is assumed to be normal for the purpose of calculating the confidence interval.

The z-table is used to find the critical value that corresponds to the desired confidence level.

Z-tables and calculator functions may present cumulative area from negative infinity to z, which needs careful interpretation.

The critical value z star is found by looking for the z that leaves 3% in the tails, corresponding to 97% coverage.

The z-table is consulted to find the z value that corresponds to 97% of the area under the curve.

The critical z value z star is determined to be 1.88 for a 94% confidence interval.

The process of finding z star involves understanding the symmetry of the normal distribution and the concept of cumulative area.

The calculated z star value is crucial for constructing the confidence interval around the sample proportion.

The importance of using the correct z-table or calculator function for accurate critical value determination is emphasized.

The final calculated z star value of 1.88 is used to complete the construction of the confidence interval.

Transcripts

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Takeaways

Q & A

What is the purpose of a one-sample z interval in statistics?

What is the significance of the 94% confidence level in this scenario?

What is the sample size in Elena's study of computer defects?

How many computers with defects were found in Elena's sample?

What is the critical value z star used for in constructing a confidence interval?

Why is it necessary to look up the critical value z star in a z-table or calculator?

What is the standard error of the statistic in the context of this script?

How is the standard error of the sample proportion calculated?

What does the z-table represent and how is it used?

How does the symmetry of the normal distribution affect the calculation of the confidence interval?

What is the critical value z star that Elena should use for her 94% confidence interval?