What is a confidence interval for the population mean ?

A confidence interval for the population mean is created using a sample of a population. It is an interval of values around the sample mean for which there is a specific probability that it contains the population mean .

What is meant by the confidence level of a confidence interval for the population mean?

The confidence level of a confidence interval for the population mean is the proportion of such confidence intervals expected to contain the population mean.

True or False? Increasing the confidence level decreases the width of the confidence interval.

False. Increasing the confidence level increases the width of the confidence interval. To have more likelihood that the interval contains the mean, the interval must be bigger.

True or False? Increasing the sample size decreases the width of the confidence interval.

True. Increasing the sample size decreases the width of the confidence interval. The bigger the sample size, the more precise the confidence interval.

IBMathsDPApplications & Interpretation (AI)HLFlashcards4. Statistics & ProbabilityCombinations of Normal Distributions & Sample Mean Distributions

Combinations of Normal Distributions & Sample Mean Distributions (DP IB Applications & Interpretation (AI))

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FrontSample Mean Distribution
True or False?
If $X_{1}, X_{2,} . . ., X_{n}$ are independent normal distributions then $a_{1} X_{1} + a_{2} X_{2} + . . . + a_{n} X_{n}$ is also a normal distribution.

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True or False?
If $X_{1}, X_{2,} . . ., X_{n}$ are independent normal distributions then $a_{1} X_{1} + a_{2} X_{2} + . . . + a_{n} X_{n}$ is also a normal distribution.
True.
If $X_{1}, X_{2,} . . ., X_{n}$ are independent normal distributions then $a_{1} X_{1} + a_{2} X_{2} + . . . + a_{n} X_{n}$ is also a normal distribution.
If $X_{1}, X_{2,} . . ., X_{n}$ are independent observations of a variable $X$ , write a formula for the distribution of the sample means, $\bar{X}$ .
If $X_{1}, X_{2,} . . ., X_{n}$ are independent observations of a variable $X$ , then the distribution of the sample means is $\bar{X} = \frac{X_{1} + X_{2} + . . . + X_{n}}{n}$ .
True or False?
For any distribution, the mean of $\bar{X}$ is equal to the mean of $X$ .
True.
For any distribution, the mean of $\bar{X}$ is equal to the mean of $X$ .
If a random sample of $n$ observations is taken from a population, $X$ , with variance $σ^{2}$ , what is the variance of $\bar{X}$ ?
If a random sample of $n$ observations is taken from a population, $X$ , with variance $σ^{2}$ , then the variance of $\bar{X}$ is $\frac{σ^{2}}{n}$ .
True or False?
For any sample size, if $X$ is normally distributed then $\bar{X}$ is also normally distributed.
True.
For any sample size, if $X$ is normally distributed then $\bar{X}$ is also normally distributed.
If a random sample of $n$ observations are taken from $X ~ N (μ, σ^{2})$ , what is the distribution of $\bar{X}$ ?
If a random sample of $n$ observations are taken from $X ~ N (μ, σ^{2})$ , then $\bar{X} ~ N (μ, \frac{σ^{2}}{n})$ .
What does the Central Limit Theorem tell you about the distribution of $\bar{X}$ ?
The Central Limit Theorem states that if a large enough sample is taken (>30), then $\bar{X}$ can be approximated by a normal distribution.
A large sample of $n$ observations are taken from the distribution $X$ with mean $μ$ and variance $σ^{2}$ , describe the distribution of $\bar{X}$ .
A large sample of $n$ observations are taken from the distribution $X$ with mean $μ$ and variance $σ^{2}$ , then by the Central Limit Theorem $\bar{X}$ is approximately $N (μ, \frac{σ^{2}}{n})$ .
True or False?
If a random sample of 40 observations is taken from $X ~ N (μ, σ^{2})$ , then the Central Limit Theorem is needed to describe $\bar{X}$ .
False.
If a random sample of 40 observations is taken from $X ~ N (μ, σ^{2})$ , then the Central Limit Theorem is not needed to describe $\bar{X}$ .
If $X$ is normally distributed, then $\bar{X}$ is also normally distributed for any sample size.
What is a confidence interval for the population mean?
A confidence interval for the population mean is created using a sample of a population.
It is an interval of values around the sample mean for which there is a specific probability that it contains the population mean.
What is meant by the confidence level of a confidence interval for the population mean?
The confidence level of a confidence interval for the population mean is the proportion of such confidence intervals expected to contain the population mean.
When should you use a normal distribution for a confidence interval (z-interval)?
You should use a normal distribution for a confidence interval (z-interval) when the population variance, $σ^{2}$ , is known.
When should you use a t-distribution for a confidence interval (t-interval)?
You should use a t-distribution for a confidence interval (t-interval) when the population variance, $σ^{2}$ , is unknown.
True or False?
Increasing the confidence level decreases the width of the confidence interval.
False.
Increasing the confidence level increases the width of the confidence interval.
To have more likelihood that the interval contains the mean, the interval must be bigger.
True or False?
Increasing the sample size decreases the width of the confidence interval.
True.
Increasing the sample size decreases the width of the confidence interval.
The bigger the sample size, the more precise the confidence interval.
It is claimed that the mean of a population is $μ_{0}$ .
A 95% confidence interval is constructed and $μ_{0}$ is outside the interval.
Is there sufficient evidence to reject the claim?
It is claimed that the mean of a population is $μ_{0}$ .
A 95% confidence interval is constructed and $μ_{0}$ is outside the interval.
There is sufficient evidence to reject the claim.
Three samples are used to create 95% confidence intervals for the population mean.
How would you find the probability that all three confidence intervals contain the population mean?
Three samples are used to create 95% confidence intervals for the population mean.
The probability that all three confidence intervals contain the population mean is $0 . 95^{3}$ .

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