Estimating the Mean in Statistics

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Questions and Answers

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?

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?

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 n ≥ 30 and normality cannot be assumed. Signup and view all the answers

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.

173.82 cm Signup and view all the answers

What is the formula to calculate the confidence interval for the population mean when the standard deviation is known?

CI = x̄ ± Zα/2 √(σ/n) Signup and view all the answers

In case II where the population standard deviation is unknown, which value do you use for the critical value?

tα/2,ν Signup and view all the answers

What does the variable ν represent in the confidence interval formula for an unknown standard deviation?

Degrees of Freedom Signup and view all the answers

Which of the following is NOT a requirement for calculating a confidence interval for the population mean?

Population standard deviation must be known Signup and view all the answers

In the example given, what is the calculated sample mean weight of the bags of rice?

5.7 kg Signup and view all the answers

Up to what degree of confidence did the example calculate the confidence interval for the bags of rice?

95% Signup and view all the answers

How is the sample standard deviation calculated in the context provided?

s = √(Σ(xi - x̄)²/(n-1)) Signup and view all the answers

What is the area remaining in each tail for the critical value Zα/2 in the confidence interval calculation?

α/2 Signup and view all the answers

What was the average score for students taught using method B?

78 Signup and view all the answers

What is the population standard deviation for method A?

10 Signup and view all the answers

What is the formula used for calculating the confidence interval for the difference in population means when population variances are known?

(x̄B − x̄A) ± zα/2(√(σA/nA) + √(σB/nB)) Signup and view all the answers

What is the z-value used for a 96% confidence level from the Z-table?

2.055 Signup and view all the answers

What is the difference in average scores between methods B and A?

8 Signup and view all the answers

Which of the following is the correct lower bound of the 96% confidence interval for µB − µA?

2.26 Signup and view all the answers

If the sample size for method B was increased to 100, how would this affect the confidence interval?

It would become narrower. Signup and view all the answers

What is the upper bound of the 96% confidence interval calculated for µB − µA?

13.74 Signup and view all the answers

What is the formula used to calculate the pooled variance?

$s^2_p = \frac{(n_1 - 1)s^2_1 + (n_2 - 1)s^2_2}{n_1 + n_2 - 2}$ Signup and view all the answers

What is the critical value $t_{\alpha/2}$ for a 95% confidence interval with 20 degrees of freedom?

2.086 Signup and view all the answers

If $\bar{x}_1 = 85.3$ and $\bar{x}_2 = 78.2$, what is $\bar{x}_1 - \bar{x}_2$?

7.1 Signup and view all the answers

What is the value of the population standard deviation estimator $s_p$?

3.97 Signup and view all the answers

What range does the 95% confidence interval for the difference between the population means $\mu_1 - \mu_2$ fall into?

$3.554 < \mu_1 - \mu_2 < 10.646$ Signup and view all the answers

How is the confidence interval for the difference calculated?

Add and subtract the standard error from the mean difference. Signup and view all the answers

What does the symbol $\nu$ represent in the context of confidence intervals?

Degrees of freedom Signup and view all the answers

What is the formula for the standard error when the population standard deviation is known?

$rac{ ext{σ}}{ ext{n}}$ Signup and view all the answers

What is the 90% confidence interval for the population mean based on the provided example?

1176.74 to 1223.26 Signup and view all the answers

In estimating the difference between two population means, what does the term $z_{α/2}$ represent?

The critical value for the normal distribution Signup and view all the answers

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$?

They allow for the assessment of the variability in the samples. Signup and view all the answers

What do the symbols $x̄_1$ and $x̄_2$ represent in the context of estimating the difference between two means?

The means of two independent random samples Signup and view all the answers

If both sample variances are known, which formula would be used to construct a confidence interval for the difference between two means?

($x̄_1 - x̄_2$) ± $z_{α/2}rac{ ext{σ}_1^2 + ext{σ}_2^2}{n_1 n_2}$ Signup and view all the answers

In the example provided, what is the number of students taught using method A?

50 Signup and view all the answers

What does the standard error represent in statistics?

An estimate of how far the sample mean is likely to be from the population mean Signup and view all the answers

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)
Standard Error: This is the standard deviation of the sample mean (X̄), 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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Description

This quiz focuses on estimating the mean using both known and unknown population standard deviations. It covers confidence intervals, the use of the t-distribution, and the concept of standard error. Test your understanding of these crucial statistical concepts with practical examples.