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Questions and Answers
What is the critical value for a 95% confidence interval with 6 degrees of freedom?
What is the critical value for a 95% confidence interval with 6 degrees of freedom?
- 2.207
- 1.943
- 2.447 (correct)
- 2.576
What would be the lower limit of the 95% confidence interval given the sample mean of 5?
What would be the lower limit of the 95% confidence interval given the sample mean of 5?
- 4.738 (correct)
- 4.700
- 4.905
- 4.808
Which formula is used when the population standard deviation is unknown for constructing a confidence interval?
Which formula is used when the population standard deviation is unknown for constructing a confidence interval?
- x̄ ± zα/2 √(n/σ)
- x̄ ± tα/2 √(n/s)
- x̄ − tα/2 √(s/n) < µ < x̄ + tα/2 √(s/n) (correct)
- x̄ − zα/2 √(σ/n) < µ < x̄ + zα/2 √(σ/n)
When is the large-sample confidence interval applicable?
When is the large-sample confidence interval applicable?
Calculate the upper limit of the 99% confidence interval for the population mean height, given a sample mean height of 170 cm and sample standard deviation of 8 cm.
Calculate the upper limit of the 99% confidence interval for the population mean height, given a sample mean height of 170 cm and sample standard deviation of 8 cm.
What is the formula to calculate the confidence interval for the population mean when the standard deviation is known?
What is the formula to calculate the confidence interval for the population mean when the standard deviation is known?
In case II where the population standard deviation is unknown, which value do you use for the critical value?
In case II where the population standard deviation is unknown, which value do you use for the critical value?
What does the variable ν represent in the confidence interval formula for an unknown standard deviation?
What does the variable ν represent in the confidence interval formula for an unknown standard deviation?
Which of the following is NOT a requirement for calculating a confidence interval for the population mean?
Which of the following is NOT a requirement for calculating a confidence interval for the population mean?
In the example given, what is the calculated sample mean weight of the bags of rice?
In the example given, what is the calculated sample mean weight of the bags of rice?
Up to what degree of confidence did the example calculate the confidence interval for the bags of rice?
Up to what degree of confidence did the example calculate the confidence interval for the bags of rice?
How is the sample standard deviation calculated in the context provided?
How is the sample standard deviation calculated in the context provided?
What is the area remaining in each tail for the critical value Zα/2 in the confidence interval calculation?
What is the area remaining in each tail for the critical value Zα/2 in the confidence interval calculation?
What was the average score for students taught using method B?
What was the average score for students taught using method B?
What is the population standard deviation for method A?
What is the population standard deviation for method A?
What is the formula used for calculating the confidence interval for the difference in population means when population variances are known?
What is the formula used for calculating the confidence interval for the difference in population means when population variances are known?
What is the z-value used for a 96% confidence level from the Z-table?
What is the z-value used for a 96% confidence level from the Z-table?
What is the difference in average scores between methods B and A?
What is the difference in average scores between methods B and A?
Which of the following is the correct lower bound of the 96% confidence interval for µB − µA?
Which of the following is the correct lower bound of the 96% confidence interval for µB − µA?
If the sample size for method B was increased to 100, how would this affect the confidence interval?
If the sample size for method B was increased to 100, how would this affect the confidence interval?
What is the upper bound of the 96% confidence interval calculated for µB − µA?
What is the upper bound of the 96% confidence interval calculated for µB − µA?
What is the formula used to calculate the pooled variance?
What is the formula used to calculate the pooled variance?
What is the critical value $t_{\alpha/2}$ for a 95% confidence interval with 20 degrees of freedom?
What is the critical value $t_{\alpha/2}$ for a 95% confidence interval with 20 degrees of freedom?
If $\bar{x}_1 = 85.3$ and $\bar{x}_2 = 78.2$, what is $\bar{x}_1 - \bar{x}_2$?
If $\bar{x}_1 = 85.3$ and $\bar{x}_2 = 78.2$, what is $\bar{x}_1 - \bar{x}_2$?
What is the value of the population standard deviation estimator $s_p$?
What is the value of the population standard deviation estimator $s_p$?
What range does the 95% confidence interval for the difference between the population means $\mu_1 - \mu_2$ fall into?
What range does the 95% confidence interval for the difference between the population means $\mu_1 - \mu_2$ fall into?
How is the confidence interval for the difference calculated?
How is the confidence interval for the difference calculated?
What does the symbol $\nu$ represent in the context of confidence intervals?
What does the symbol $\nu$ represent in the context of confidence intervals?
What is the formula for the standard error when the population standard deviation is known?
What is the formula for the standard error when the population standard deviation is known?
What is the 90% confidence interval for the population mean based on the provided example?
What is the 90% confidence interval for the population mean based on the provided example?
In estimating the difference between two population means, what does the term $z_{α/2}$ represent?
In estimating the difference between two population means, what does the term $z_{α/2}$ represent?
When computing a confidence interval for the difference between two means, what is the role of the variances $ ext{σ}^2_1$ and $ ext{σ}^2_2$?
When computing a confidence interval for the difference between two means, what is the role of the variances $ ext{σ}^2_1$ and $ ext{σ}^2_2$?
What do the symbols $x̄_1$ and $x̄_2$ represent in the context of estimating the difference between two means?
What do the symbols $x̄_1$ and $x̄_2$ represent in the context of estimating the difference between two means?
If both sample variances are known, which formula would be used to construct a confidence interval for the difference between two means?
If both sample variances are known, which formula would be used to construct a confidence interval for the difference between two means?
In the example provided, what is the number of students taught using method A?
In the example provided, what is the number of students taught using method A?
What does the standard error represent in statistics?
What does the standard error represent in statistics?
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Study Notes
Estimating the Mean (µ)
- Single Sample:
- Population Standard Deviation (σ) known: The confidence interval (CI) is calculated using the formula:
- CI = x̄ ± Zα/2 (σ/√n)
- Where: x̄ is the sample mean, Zα/2 is the critical value from the standard normal distribution, σ is the known population standard deviation, and n is the sample size.
- Population Standard Deviation (σ) unknown: The confidence interval is calculated using the t-distribution:
- CI = x̄ ± tα/2 (ν) (s/√n)
- Where: x̄ is the sample mean, s is the sample standard deviation, tα/2,ν is the critical value from the t-distribution with n-1 degrees of freedom, and α is the significance level.
- Large Sample Size (n≥30): For non-normal populations, the confidence interval formula is similar to the known σ case, but σ can be replaced by s:
- CI = x̄ ± zα/2 (σ/√n) or CI = x̄ ± zα/2 (s/√n)
- Population Standard Deviation (σ) known: The confidence interval (CI) is calculated using the formula:
- Standard Error: This is the standard deviation of the sample mean (XÌ„), and it is calculated as follows:
- Known σ: Standard Error = σ/√n
- Unknown σ: Standard Error = s/√n
Two Samples: Estimating the Difference Between Two Means (µ1 - µ2)
- Population Standard Deviations (σ1, σ2) known: The confidence interval for the difference between two population means is calculated using the following formula:
- CI = (x̄1 - x̄2) ± zα/2 √(σ1²/n1 + σ2²/n2)
- Population Standard Deviations (σ1, σ2) unknown: The confidence interval is calculated using the t-distribution and pooled standard deviation:
- CI = (x̄1 - x̄2) ± tα/2 (ν) sp √(1/n1 + 1/n2)
- Where: sp is the pooled standard deviation, calculated as √((n1-1)s1² + (n2-1)s2² / (n1 + n2 - 2)), and ν (degrees of freedom) is n1 + n2 - 2.
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