Question

a) Use software to determine how large a sample size n is needed for the critical value of the t distribution to be within 0.01 of the corresponding critical value of the Normal distribution for a 90%, 95%, and 99% confidence interval for a population mean. (Enter your answers as whole numbers.) for 90%, n= for 95%, n= for 99%, n=

Answer 1

To determine the sample size where the t-distribution critical values are within 0.01 of the normal distribution critical values for different confidence intervals, one would need to use statistical software. The sample sizes are related to the degrees of freedom, which approach the z-scores of 1.645, 1.96, and 2.576 for the 90%, 95%, and 99% confidence intervals respectively.

Step-by-step explanation:

The sample size required for the critical values of the t-distribution to be within 0.01 of the z-scores can be determined using statistical software. These calculations are based on the desired confidence level and the degrees of freedom, which are directly related to the sample size.

For Each Confidence Level:

• 90% Confidence Interval: Use a z-score of 1.645 as the critical value from the normal distribution.
• 95% Confidence Interval: The critical value from the normal distribution is 1.96 corresponding to a z-score that places 2.5% in each tail of the distribution.
• 99% Confidence Interval: The critical value from the normal distribution is 2.576.

Since we are seeking a sample size where the t-distribution nearly matches the normal distribution, we are looking for a case where the sample size is large enough that the degrees of freedom yields a t-score that is within 0.01 of those z-scores. To perform these calculations, statistical software or a graphing calculator with statistical functions would be used. The exact numbers would depend on the software's algorithm for determining degrees of freedom where the t-distribution approaches the normal distribution.

Typically, for large sample sizes (usually greater than 30), the t-distribution approaches the normal distribution. However, without performing the actual calculation with software, we cannot provide exact sample sizes for this query.

Answer 2

The question involves finding the sample size needed for the t-distribution critical value to approximate that of the normal distribution for 90%, 95%, and 99% confidence intervals in estimating a population mean. Statistical software would be needed to calculate these specific sample sizes.

Step-by-step explanation:

The question seeks to determine sample size for achieving a t-distribution critical value that's within 0.01 of the normal distribution's critical value for different confidence levels for estimating a population mean.

For a 90% confidence interval, we want to capture the central 90% of the probability distribution. Using a z-score for a normal distribution, the critical value is 1.645. Using a t-distribution, we must determine the sample size where the critical t-value is within 0.01 of 1.645.

For 95% and 99% confidence intervals, the process is similar. First, we establish the normal distribution's z-scores for 95% (1.96) and 99% (2.576) confidence intervals. Then, we look for the respective t-distribution sample sizes where the critical t-values approximate the z-scores within 0.01.

Since the question requires the use of software to find the exact sample sizes, we must rely on statistical software or functions within a graphing calculator to derive these values. As we do not perform actual software calculations here, we cannot provide specific values, but the method involves inputting the confidence level and the desired proximity to the z-score to find the necessary sample size, which usually increases with a narrower gap between the t-value and the z-score.

It's important to note that historically, a rule of thumb was to use the Student's t-distribution for sample sizes up to 30 and the normal approximation for larger samples. However, with modern technology, the t-distribution is usually preferred when the population standard deviation is unknown, regardless of the sample size.