Statistics Chapter 8: Estimates and Sample Sizes

<p>2070 (B)</p> Signup and view all the answers

<p>To separate likely from unlikely sample means (B)</p> Signup and view all the answers

<p>When the sample is a simple random sample (A)</p> Signup and view all the answers

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

<p>The probability of falling in the right tail of the distribution (C)</p> Signup and view all the answers

<p>The sample size must be equal to the population size (D)</p> Signup and view all the answers

<p>24 (D)</p> Signup and view all the answers

<p>$300 (C)</p> Signup and view all the answers

<p>$4.95 (D)</p> Signup and view all the answers

<p>2.66 (C)</p> Signup and view all the answers

<p>181.05 to 190.95 (A)</p> Signup and view all the answers

<p>x̄ ± tα/2 (s/√n) (B)</p> Signup and view all the answers

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

<p>0.05 (C)</p> Signup and view all the answers

<p>The margin of error is $178.02$. (A), There is a 95% probability that the population mean falls within this interval. (B)</p> Signup and view all the answers

<p>n = $rac{h z_{rac{α}{2}} σ^2}{E^2}$ (D)</p> Signup and view all the answers

<p>1.96 (D)</p> Signup and view all the answers

<p>Specific conditions of the sample must be met. (B)</p> Signup and view all the answers

<p>The margin of error (A)</p> Signup and view all the answers

<p>It provides a 95% confidence that the sample mean lies within 3 IQ points from the population mean. (B)</p> Signup and view all the answers

<p>$178.02$ (A)</p> Signup and view all the answers

<p>The sample mean is the midpoint of the confidence interval. (B), The sample mean ranges from $ar{x} - E$ to $ar{x} + E$. (D)</p> Signup and view all the answers

<p>$90.50 (C)</p> Signup and view all the answers

<p>E = zα/2 * (σ / √n) (C)</p> Signup and view all the answers

<p>1.65 (A)</p> Signup and view all the answers

<p>It denotes the probability of making a Type I error. (D)</p> Signup and view all the answers

<p>($87.95, $92.05) (A)</p> Signup and view all the answers

<p>E increases. (C)</p> Signup and view all the answers

<p>The population is normal. (A)</p> Signup and view all the answers

<p>To represent the critical value associated with the confidence level. (C)</p> Signup and view all the answers

<p>Sample size is large and population standard deviation is not known (B)</p> Signup and view all the answers

<p>T distribution is shorter and wider than standard normal distribution (D)</p> Signup and view all the answers

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

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

<p>The sample mean is the best point estimate of the population mean (B)</p> Signup and view all the answers

<p>If the population is normally distributed (D)</p> Signup and view all the answers

<p>Use the t distribution for small samples or large samples if σ is unknown (C)</p> Signup and view all the answers

<p>It becomes identical to the standard normal distribution (A)</p> Signup and view all the answers

The value of the z-score that separates the right-tail area of α/2 in the standard normal distribution.

The best point estimate of the population mean (µ) is the sample mean (x̄).

To estimate the population mean (µ) when the population standard deviation (σ) is known, we need a sample mean (x̄) and a margin of error (E).

The margin of error (E) in estimating the population mean (µ) is the maximum likely difference between the sample mean (x̄) and the population mean (µ).

Assumptions for estimating the population mean when the population standard deviation (σ) is known include a simple random sample, known σ, and either a normal population distribution or a sample size (n) greater than 30.

A range of values within which the true population mean is likely to fall, based on a sample mean and the standard deviation of the population.

The sample mean (x̄) is the best estimate of the population mean (µ). The confidence interval is a range around x̄ that represents our estimated range for µ.

The standard deviation of the population (σ) is a measure of how spread out the data is. It is used along with the sample mean (x̄) and the confidence level to determine the margin of error for the confidence interval.

The confidence level determines the percentage of confidence that the true population mean falls within the confidence interval. For a 99% confidence level, we are 99% confident that the true mean lies within the calculated range.

A numerical value that represents the best guess for the unknown population parameter. It's taken from the sample data.

A range of values that is likely to contain the true population parameter with a certain level of confidence.

The standard deviation of the population, which is assumed to be known in this context.

The sample size, which is the number of observations in the sample.

The critical value from the standard normal distribution corresponding to the desired confidence level.

The formula used to calculate the confidence interval for the population mean when the population standard deviation is known.

The population mean (µ) is the average value of a variable for the entire population.

The sample mean (x̄) is the average value of a variable calculated from a sample of data.

The z-score (zα/2) is a value from the standard normal distribution that corresponds to a specific confidence level.

The formula for calculating the sample size (n) for a known population standard deviation (σ)

The t-distribution is a bell-shaped distribution used for estimating population means when the population standard deviation is unknown. It is wider and shorter than the standard normal distribution.

The t-distribution has only one parameter: degrees of freedom (df). This value represents the number of independent pieces of information in the sample, calculated as df = n - 1.

The t-distribution approaches the standard normal distribution as the sample size (n) increases. This means the t-distribution becomes more similar to the standard normal distribution.

The sample mean (x̄) is the best single value estimate for the population mean (µ) when the population standard deviation (σ) is unknown.

When estimating a population mean with an unknown population standard deviation, a t-distribution is used for confidence intervals if the sample size is greater than or equal to 30.

When estimating a population mean with an unknown population standard deviation, a t-distribution is used for confidence intervals if the sample size is less than 30 and the population is normally distributed.

If the population is not normally distributed (or its distribution is unknown), and the population standard deviation is unknown, then a nonparametric method is used to create the confidence interval for the population mean.

The t-distribution is essential for many statistical hypothesis tests, particularly when dealing with small sample sizes and unknown population standard deviations. It plays a significant role in determining if there is a statistically significant difference between two groups or between a sample and a known population.

The degrees of freedom (df) for a t-distribution are calculated by subtracting 1 from the sample size (n). It indicates the number of independent pieces of information used to estimate the population parameter.

A confidence interval for the population mean (µ) using the t-distribution provides a range of values within which we are confident the true population mean lies. It is calculated using the sample mean (x̄), the t-critical value, the sample standard deviation (s), and the sample size (n).

The t-critical value, denoted as tα/2, is a value from the t-distribution that corresponds to the desired level of confidence and degrees of freedom. It separates the rejection region from the non-rejection region in a hypothesis test.

The margin of error (E) for a confidence interval of µ using the t-distribution is the maximum likely difference between the sample mean (x̄) and the true population mean (µ). It is calculated by multiplying the t-critical value by the standard error of the mean.

The standard error of the mean (SEM) for a t-distribution is an estimate of the standard deviation of the sampling distribution of the sample mean. It measures the variability of sample means around the population mean.

The formula for calculating the confidence interval for the population mean (µ) using the t-distribution is: x̄ ± tα/2 * (s/√n). This provides a range within which the true population mean is likely to fall with a certain degree of confidence.

The t-distribution table is used to find the critical value (tα/2) associated with a specific confidence level and degrees of freedom. The table provides t-scores for different tail probabilities and degrees of freedom, allowing you to find the appropriate t-critical value for your confidence interval.