Statistics Confidence Intervals Quiz
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Questions and Answers

Which of the following actions can reduce the margin of error when estimating a population mean?

  • Decrease the population standard deviation (correct)
  • Decrease the sample size
  • Increase the sample size (correct)
  • Increase the confidence level

What condition is necessary when using a t critical value to calculate a confidence interval?

  • Population standard deviation must be known
  • Sample must be randomly selected (correct)
  • Sample must have a normal distribution (correct)
  • Sample size must be greater than 30

Which of the following statements about sample size and confidence intervals is true?

  • Larger sample sizes increase the width of the confidence interval
  • Sample size has no effect on the confidence level
  • Increasing sample size decreases the confidence interval width (correct)
  • Smaller sample sizes lead to more reliable estimates

Which of the following is NOT a condition for using a t test?

<p>The population standard deviation must be known (D)</p> Signup and view all the answers

In what scenario would you reduce the confidence level when estimating a population mean?

<p>When you want a narrower confidence interval (A)</p> Signup and view all the answers

What does a standard deviation of 0.13 indicate compared to a standard deviation of 0.067?

<p>Higher sample-to-sample variability (B)</p> Signup and view all the answers

For a standard deviation of 0.13, how does it relate to the sample size of 50?

<p>Indicates a smaller sample size than 50 (D)</p> Signup and view all the answers

When transitioning from a sample size of n = 20 to n = 100, what happens to the standard deviation of the sampling distribution?

<p>It decreases (A)</p> Signup and view all the answers

Which statement correctly reflects how the sampling distribution is affected by sample size?

<p>A larger sample size leads to a better approximation of normality (B)</p> Signup and view all the answers

If the standard deviation for a sample with a size of n = 50 is 0.067, what does a standard deviation of 0.67 imply about the sample size?

<p>Sample size is smaller than 50 (C)</p> Signup and view all the answers

What does a population proportion of p = 0.88 suggest about the standard deviation when compared to lower proportions?

<p>Lower standard deviation (A)</p> Signup and view all the answers

Which of the following statements is true regarding sample sizes n = 20 and n = 100?

<p>Only n = 100 results in an approximately normal distribution (B)</p> Signup and view all the answers

How does the standard deviation impact the reliability of sample proportions?

<p>Lower standard deviation leads to more reliable sample proportions (D)</p> Signup and view all the answers

What is the mean (mu_(p hat)) for a candidate favored by 33.0% of registered voters?

<p>0.33 (A)</p> Signup and view all the answers

What is the standard deviation (sigma_(p hat)) for a sampling distribution if the proportion of sunny days is 0.550 and the sample size is 12?

<p>0.143 (A)</p> Signup and view all the answers

Which option correctly identifies both the mean and standard deviation for a scenario where the candidate is favored by 33.0% of voters?

<p>Mean = 33.0%, Standard deviation = 0.061 (D)</p> Signup and view all the answers

If the standard deviation (sigma_(p hat)) in the situation with 33.0% voter support is calculated as 0.061, what is the formula used to derive this value?

<p>sqrt[(p(1-p)/n)] (A)</p> Signup and view all the answers

For a candidate with a mean (mu_(p hat)) of 0.33, how would the sampling distribution look if the sample size were increased?

<p>The mean would remain at 0.33. (A)</p> Signup and view all the answers

What would be the mean (mu_(p hat)) for a situation where the proportion of sunny days is known to be 0.550?

<p>0.550 (D)</p> Signup and view all the answers

What is the standard deviation (sigma_(p hat)) for a polling organization sampling 60 voters, given a mean (mu_(p hat)) of 0.50?

<p>0.075 (B)</p> Signup and view all the answers

Which option would be incorrect regarding the standard deviation of the sampling distribution of p hat for a candidate favored by 33.0%?

<p>The correct standard deviation value is 0.272. (A)</p> Signup and view all the answers

What does a z-statistic of 4.00 indicate about the sample proportion p hat with respect to the null hypothesis value of p = 0.35?

<p>The sample proportion is 4.00 standard errors above 0.35. (D)</p> Signup and view all the answers

Which statement correctly describes the distance between the sample statistic p hat and the null hypothesis value?

<p>It is defined in relation to the standard error of p hat. (B)</p> Signup and view all the answers

In hypothesis testing, what must be considered when determining the distance between the sample statistic p hat and the population proportion p?

<p>The sample statistic must be assessed against the standard error. (B)</p> Signup and view all the answers

Which of the following distances is used in hypothesis testing to compare the sample statistic to the predicted population proportion under the alternative hypothesis?

<p>Distance in terms of the standard error of p hat. (A)</p> Signup and view all the answers

Which of the following statements must be true for a sample to be considered representative of a population?

<p>The sample must be a random selection from the population. (D)</p> Signup and view all the answers

What could indicate that the sample statistic p hat is greatly differing from the null hypothesis?

<p>A large <em>z</em>-statistic indicating multiple standard errors. (A)</p> Signup and view all the answers

What is a condition that must be met when using a large-sample confidence interval to estimate the population proportion?

<p>The sample must contain at least 10 successes and 10 failures. (D)</p> Signup and view all the answers

If you have a sample with proportion p hat = 0.65, which condition must be satisfied for the confidence interval calculation?

<p>The product of n times p hat must be at least 10. (D)</p> Signup and view all the answers

For a sample size of 100 with a proportion p hat of 0.25, can a large-sample 95% confidence interval be used?

<p>Yes, because n times p hat is sufficient. (D)</p> Signup and view all the answers

Which statement regarding the population proportion estimation is incorrect?

<p>The sample size can be arbitrarily small. (C)</p> Signup and view all the answers

What is the minimum requirement for successes in a sample to apply a large-sample confidence interval?

<p>At least 10 successes. (C)</p> Signup and view all the answers

Which of the following best describes the necessary conditions for using a confidence interval based on the normal approximation?

<p>Both n times p hat and n times (1 - p hat) must be greater than or equal to 10. (C)</p> Signup and view all the answers

Which of the following is NOT a requirement for a valid random sample?

<p>The sample must include diverse demographics. (C)</p> Signup and view all the answers

What is a sampling distribution?

<p>The distribution of the possible values of a statistic. (A)</p> Signup and view all the answers

In which scenario can we say that the sampling distribution of x bar will be approximately normal?

<p>Small samples from an approximately normal population. (A), Large samples from an approximately normal population. (C)</p> Signup and view all the answers

Which of the following statements about sampling distributions is false?

<p>The average value of a sampling distribution is always higher than the population parameter. (B)</p> Signup and view all the answers

Which category describes large samples from a population that is not approximately normal?

<p>The sampling distribution may become approximately normal according to the Central Limit Theorem. (C)</p> Signup and view all the answers

What determines the shape of a sampling distribution?

<p>Both the size of the sample and the shape of the population distribution. (D)</p> Signup and view all the answers

For small samples taken from populations that are not approximately normal, what can be expected regarding the shape of the sampling distribution?

<p>The sampling distribution will be skewed. (C)</p> Signup and view all the answers

Which situation leads to the most reliable sampling distribution?

<p>Large samples from an approximately normal population. (C)</p> Signup and view all the answers

What will the distribution of possible values of a statistic depend on when it comes to sample size and population distribution?

<p>The variability of the population and the sample size. (A)</p> Signup and view all the answers

Flashcards

Mean of Sampling Distribution (µ̂)

The average value of the sample proportions (p̂) that would be obtained from many samples of a given size, drawn from a population.

Standard Deviation of Sampling Distribution (σ̂)

A measure of the variability or spread of sample proportions (p̂) around the mean of the sampling distribution. Smaller standard deviation signifies less variation.

Sampling Distribution

A distribution of sample statistics obtained from a large number of samples of the same size taken from the same population.

Sample Proportion (p̂)

The proportion of successes (or a specific characteristic) in a given sample.

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Population proportion (p)

The proportion of a characteristic within an entire population.

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Sample Size (n)

The number of individuals or items included in a sample.

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Standard Deviation formula (p̂)

Standard deviation of sampling distribution of p̂ = √[(p(1-p))/n)]

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Central Limit Theorem

The theorem that describes how the sample distribution becomes approximately normal as the sample size increases.

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Sample size vs. Variability

A larger sample size generally results in lower sample-to-sample variability in sample proportions.

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Standard deviation of p-hat

Standard deviation of the sampling distribution of the sample proportion, sigma_(p hat).

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Sample size and variability (SD = 0.13)

A sample size resulting in a standard deviation of 0.13 for the sampling distribution of sample proportions is smaller than for a size of 50 if the standard deviation for n=50 is smaller (like 0.067).

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Normal approximation

When the sample size (n) is sufficiently large, the sampling distribution of the sample proportion (p-hat) is approximately normal.

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Population proportion (p)= 0.88, Sample size

For a population proportion of 0.88 , a larger sample size (n=100) will have a sampling distribution with a lower variability (smaller standard deviation) of p than a smaller sample size (n=20).

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Sampling Distribution normality

The sampling distribution of the sample proportion is approximately normal when the sample size is large enough (a rule of thumb is np and n(1-p) are both greater than or equal to 10).

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Sample Size Effect on Variability (n=20 vs. n=100)

A larger sample size (like n= 100) results in a sampling distribution with less variability (lower standard deviation), compared to a smaller sample size (like n= 20).

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Standard Error

Standard deviation of a sampling distribution of a particular statistic (or in this case, the sample proportion p).

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Large Sample Size

A sample size considered sufficiently large (usually n ≥ 30) to allow the Central Limit Theorem to apply. This means the sampling distribution of the sample mean will be approximately normal.

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Small Sample Size

A sample size considered too small (usually n < 30) to assume the sampling distribution of the sample mean will be normal. The shape of the sampling distribution will depend on the shape of the population distribution.

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Approximately Normal Population

A population distribution that resembles a bell-shaped curve, where values are clustered around the mean.

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Non-Normal Population

A population distribution that does not resemble a bell curve, might be skewed, or have multiple peaks.

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Shape of Sampling Distribution (Small Sample, Normal Pop)

For small samples from a normal population, the sampling distribution of the sample mean will also be approximately normal.

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Shape of Sampling Distribution (Large Sample, Non-Normal Pop)

For large samples from a non-normal population, the sampling distribution of the sample mean will be approximately normal, thanks to the Central Limit Theorem.

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Shape of Sampling Distribution (Small Sample, Non-Normal Pop)

For small samples from a non-normal population, the shape of the sampling distribution of the sample mean is unknown and cannot be assumed to be normal.

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Z-statistic

A standardized score that measures how many standard errors a sample statistic (like the sample proportion) is away from the null hypothesis value.

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Standard Error of the Sample Proportion

A measure of the variability or spread of sample proportions around the population proportion. It estimates the spread of the sampling distribution of sample proportions.

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Hypothesis Testing

A statistical procedure used to determine whether there is enough evidence to reject the null hypothesis.

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Null Hypothesis

A statement about a population parameter that is assumed to be true until evidence suggests otherwise.

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Alternative Hypothesis

A statement that contradicts the null hypothesis and represents the alternative explanation for the observed data.

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Confidence Interval & Sample Size

Increasing the sample size results in a narrower confidence interval, leading to a more precise estimate of the population mean. This is because a larger sample size reduces the standard error of the mean.

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Confidence Interval & Confidence Level

Decreasing the confidence level makes the confidence interval narrower. This means you are less certain about the population mean, but you get a more precise estimate. A 90% confidence interval is narrower than a 95% interval.

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t-Distribution: Sample Size

When calculating a confidence interval for a population mean using the t-distribution, a sample size of 30 or less is required. This is because the t-distribution assumes a normal distribution, and small sample sizes make this assumption less accurate.

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t-Distribution & Random Sampling

For a valid confidence interval using the t-distribution, the sample must be randomly selected or represent the population accurately. This ensures that the sample is free from bias and accurately reflects the characteristics of the population.

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t-Distribution: Population Standard Deviation

When using the t-distribution to calculate a confidence interval for a population mean, you don't need to know the population standard deviation. This is one of the key advantages of the t-distribution.

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Random Sampling

Selecting a sample from a population in a way that gives each individual an equal chance of being chosen. This ensures the sample is representative of the overall population.

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Sample Size

The number of individuals or items included in a sample. A larger sample size generally leads to more accurate results.

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Successes & Failures

When applying a confidence interval for proportions, both the number of successes and failures in the sample must be sufficiently large (at least 10 each) for valid results.

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Large-Sample Confidence Interval

A method used to estimate a population proportion with a certain level of confidence. Requires a large sample size and certain conditions to be met.

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Conditions for Large-Sample Confidence Interval

To use a large-sample confidence interval for proportions, the following conditions must be met: 1. Random sampling, 2. At least 10 successes and 10 failures in the sample

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Sample Proportion

The proportion of successes (or a specific characteristic) observed in a given sample. It is denoted as p̂.

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n p hat

The product of the sample size (n) and the sample proportion (p̂). It represents the expected number of successes in the sample. For a large-sample confidence interval, this product should be at least 10.

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n (1 - p hat)

The product of the sample size (n) and (1 minus the sample proportion (p̂)). It represents the expected number of failures in the sample. For a large-sample confidence interval, this product should also be at least 10.

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Study Notes

Sampling Distribution

  • A sampling distribution displays the distribution of possible values for a statistic calculated from samples.
  • It helps determine the expected value and variability for a statistic in a population.
  • Sampling variability is the variability of a statistic from sample to sample.

Sampling Variability

  • Varies with sample size.
  • Not constant across samples.
  • Describes the difference between the observed value of a statistic and the population parameter.
  • Shows the differences in values, calculated from samples of four.

Characteristics of Sampling Distribution

  • Becomes less spread out as sample size increases.
  • Not affected by changes in sample size.
  • Becomes less spread out when the sample proportion is 0.5.

Sample Size and Standard Deviation

  • Sample size increases, standard deviation decreases (of the sampling distribution).
  • Larger sample sizes result in a narrower sampling distribution.

Population Proportion and Sampling Distribution

  • Sampling distribution is approximately normal when the sample size is large and the proportion (p) is not close to 0 or 1, and the sample size multiplies (np) and n(1-p) are both greater than or equal to 10.
  • Sampling distribution's average is equal to the population proportion.
  • Standard deviation of the sampling distribution depends on sample proportion and sample size (np and n(1-p)).

Statistics and Sampling Distribution

  • The sampling distribution of the statistic has a spread (or standard deviation), that is dependent on sample size and population proportion, for large samples.
  • The spread (or standard deviation) of the sampling distribution depends on characteristics like population mean, standard deviation and sample size.

Confidence Intervals for Population Proportion

  • Sample size, confidence level, and sample proportion all effect the width of a confidence interval.
  • Margin of error is related to the sample size and variability.

Hypothesis Testing

  • Fail to reject the null hypothesis if the data does not provide convincing evidence against it.
  • Reject the null hypothesis if the data provides convincing evidence against it.
  • Initially assume the null hypothesis is true.

Hypothesis Testing and Errors

  • Type I error is rejecting a true null hypothesis.
  • Type II error is failing to reject a false null hypothesis.
  • Significance level affects the probability of rejecting a null hypothesis when it's true.

Paired Samples t-tests

  • Used in studies where sample sizes can be large, sample differences can be treated as a random sample, and the population of differences is approximately normal.

Hypothesis Testing for Two Populations

  • Ensure sample size allows for at least 10 successes and 10 failures in each group.
  • Appropriate when samples are random.

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Description

Test your understanding of confidence intervals and t tests with this quiz. Explore essential conditions, actions to reduce margin of error, and truths about sample size. Challenge your knowledge about statistical estimation techniques.

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