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?

The population standard deviation must be known Signup and view all the answers

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

When you want a narrower confidence interval Signup and view all the answers

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

Higher sample-to-sample variability Signup and view all the answers

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

Indicates a smaller sample size than 50 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?

It decreases Signup and view all the answers

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

A larger sample size leads to a better approximation of normality 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?

Sample size is smaller than 50 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?

Lower standard deviation Signup and view all the answers

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

Only n = 100 results in an approximately normal distribution Signup and view all the answers

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

Lower standard deviation leads to more reliable sample proportions Signup and view all the answers

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

0.33 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?

0.143 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?

Mean = 33.0%, Standard deviation = 0.061 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?

sqrt[(p(1-p)/n)] 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?

The mean would remain at 0.33. 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?

0.550 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?

0.075 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%?

The correct standard deviation value is 0.272. 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?

The sample proportion is 4.00 standard errors above 0.35. Signup and view all the answers

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

It is defined in relation to the standard error of p hat. 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?

The sample statistic must be assessed against the standard error. 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?

Distance in terms of the standard error of p hat. Signup and view all the answers

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

The sample must be a random selection from the population. Signup and view all the answers

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

A large z-statistic indicating multiple standard errors. 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?

The sample must contain at least 10 successes and 10 failures. 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?

The product of n times p hat must be at least 10. 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?

Yes, because n times p hat is sufficient. Signup and view all the answers

Which statement regarding the population proportion estimation is incorrect?

The sample size can be arbitrarily small. Signup and view all the answers

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

At least 10 successes. 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?

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

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

The sample must include diverse demographics. Signup and view all the answers

What is a sampling distribution?

The distribution of the possible values of a statistic. Signup and view all the answers

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

Small samples from an approximately normal population. Signup and view all the answers

Which of the following statements about sampling distributions is false?

The average value of a sampling distribution is always higher than the population parameter. Signup and view all the answers

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

The sampling distribution may become approximately normal according to the Central Limit Theorem. Signup and view all the answers

What determines the shape of a sampling distribution?

Both the size of the sample and the shape of the population distribution. 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?

The sampling distribution will be skewed. Signup and view all the answers

Which situation leads to the most reliable sampling distribution?

Large samples from an approximately normal population. 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?

The variability of the population and the sample size. Signup and view all the answers

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.