Assignment III - PART B - Answers

 

PART B 

1. Steps Involved in Hypothesis Testing:

There are typically six steps involved in hypothesis testing:

1. State the Hypotheses:

   • Null Hypothesis (H₀): This is the default statement, often assuming no difference or effect.
   • Alternative Hypothesis (H₁): This is the opposite of the null hypothesis, proposing a difference or effect.

2. Set the Significance Level (α): This is the probability of rejecting the null hypothesis when it's actually true (type I error). Common choices are 0.05 (5%) or 0.01 (1%).

3. Select the Test Statistic: This is a statistical measure used to assess the evidence against the null hypothesis. The choice depends on factors like sample size and data type.

4. Determine the Decision Rule: Based on the significance level and test statistic, you define a critical region (rejection zone). If the test statistic falls within this region, you reject the null hypothesis.

5. Collect and Analyze Data: You gather your sample data and calculate the test statistic.

6. Interpret the Results: Based on the decision rule, you either reject the null hypothesis (evidence suggests a difference) or fail to reject it (insufficient evidence for a difference).

2. Hypothesis Testing Example:

Given: Population Mean (μ) = 0.700, Sample Mean (x̄) = 0.742, Sample Standard Deviation (s) = 0.040, Sample Size (n) = 10.

To test the null hypothesis (H₀: μ = 0.700), we could use a one-sample Z-test. However, for 3 marks, you can simply state the approach:

• Calculate the test statistic (e.g., Z-score).
• Determine the critical value for your chosen significance level (α).
• Compare the test statistic to the critical value.
• If the test statistic falls outside the critical region (far from μ), reject H₀. Otherwise, fail to reject H₀.

Work out :

Testing the Null Hypothesis (One-Sample Z-Test)

We can use a one-sample Z-test to evaluate the null hypothesis (H₀: μ = 0.700) for the population mean, where μ is the actual population mean. Here's the step-by-step solution:

1. Calculate the Z-score:

   The Z-score measures how many standard deviations the sample mean (x̄) deviates from the hypothesized population mean (μ).

   Z = (x̄ - μ) / (s / √n)

   where:

   • x̄ = Sample mean (0.742)
   • μ = Hypothesized population mean (0.700)
   • s = Sample standard deviation (0.040)
   • n = Sample size (10)

   Z = (0.742 - 0.700) / (0.040 / √10) Z ≈ 1.05

2. Choose the Significance Level (α):

   Let's assume a common significance level of α = 0.05 (5%).

3. Determine the Critical Z-value:

   For a two-tailed test (alternative hypothesis: not equal to), we need to consider both positive and negative critical values at α/2 = 0.025 (2.5%) on either tail of the standard normal distribution.

   You can use a Z-table or software to find the critical Z-values. In this case, the critical Z-values are approximately ±1.96.

4. Decision Rule:

   • Reject H₀: If the calculated Z-score falls outside the critical region (<-1.96 or >1.96).
   • Fail to Reject H₀: If the Z-score falls within the critical region (-1.96 to 1.96).

5. Interpretation:

   Our calculated Z-score (1.05) falls within the critical region (-1.96 to 1.96). Based on the decision rule, we fail to reject the null hypothesis (H₀: μ = 0.700) at the 5% significance level.

Conclusion:

With the available information, we don't have sufficient evidence to reject the claim that the population mean is 0.700. However, it's important to consider limitations:

• A sample size of 10 is relatively small. Larger samples might provide more conclusive results.
• We haven't explored the p-value, which gives the exact probability of observing a Z-score as extreme or more extreme than the calculated value, assuming the null hypothesis is true.

For a more comprehensive analysis, consider using statistical software to calculate the p-value and explore the impact of different significance levels.

3. Completely Randomized Design (CRD):

A CRD is an experimental design where treatments are randomly assigned to experimental units. This randomization helps control for extraneous variables and ensures unbiased estimates of treatment effects.

Key features:

• Treatments are randomly assigned to subjects or groups.
• Each subject or group has an equal chance of receiving any treatment.
• Subjects or groups are independent of each other.

4. Confidence Interval for Standard Deviation:

Absolutely, here's how to find the 95% confidence limits for the standard deviation of the electric bulb lights, given a sample standard deviation of 100 hours and a sample size of 200:

1. Chi-Square Distribution: We'll use the chi-square (χ²) distribution to find the confidence limits. This distribution is used for estimating population standard deviation based on sample standard deviation.

2. Confidence Level and Degrees of Freedom:

   • We want a 95% confidence level. This translates to a 2.5% area in each tail of the chi-square distribution (since 100% - 95% = 5%, and half goes in each tail).

   • Degrees of freedom (df) for this problem are n-1, where n is the sample size. df = 200 (sample size) - 1 = 199

3. Look Up Chi-Square Values:

   • We need two chi-square values:

     • Lower chi-square value (χ²lower) with 199 degrees of freedom and a cumulative area of 0.025 (2.5% in the lower tail).
     • Upper chi-square value (χ²upper) with 199 degrees of freedom and a cumulative area of 0.975 (95% + 2.5% in the lower tail).

   • You can find these values using a chi-square table or statistical software.

4. Confidence Limits Formula:

   • The confidence limits for the population standard deviation (σ) are:

     Lower Limit (LL) = s * sqrt( χ²lower / df ) Upper Limit (UL) = s * sqrt( χ²upper / df )

   where:

   • s is the sample standard deviation (100 hours in this case).
   • χ²lower and χ²upper are the chi-square values found in step 3.
   • df is the degrees of freedom (199).

5. Calculate the Confidence Limits:

   • Once you have the chi-square values, plug them into the formula along with s and df.
   • The calculation will give you the lower and upper confidence limits for the population standard deviation of the electric bulb lifetimes.

Note: You'll need to refer to a chi-square table or software to find the specific chi-square values for the given degrees of freedom and cumulative areas.

5. Properties of Estimators:

An estimator is a statistic used to estimate an unknown population parameter. Here are some important properties:

• Unbiasedness: An unbiased estimator, on average, estimates the true parameter accurately. The average of many estimates should be close to the parameter. (Example: Sample mean is an unbiased estimator of population mean)
• Consistency: As the sample size increases, a consistent estimator gets closer and closer to the true parameter value. (Example: Sample proportion is a consistent estimator of population proportion)
• Efficiency: Among unbiased estimators, the efficient one has the smallest variance (less spread around the true value). (Example: Sample mean is generally more efficient than the median)

6. Hypothesis Testing Concepts:

Hypothesis testing is a statistical method to assess claims about a population parameter. Here's a brief explanation:

• We formulate two competing hypotheses: null hypothesis (H₀) stating no effect or difference, and alternative hypothesis (H₁) proposing an effect or difference.
• We choose a significance level (α) to control the risk of wrongly rejecting a true null hypothesis.
• We select a test statistic based on the data and hypotheses.
• We define a decision rule based on the significance level and test statistic.
• We analyze the data and compare the test statistic to the decision rule.
• Based on the comparison, we either reject the null hypothesis (evidence suggests a difference) or fail to reject it (insufficient evidence).

7. Uses of ANOVA Test:

ANOVA (Analysis of Variance) is a statistical technique used to compare the means of several groups. Here are some common uses:

• Comparing the effectiveness of different treatments or interventions.
• Analyzing the impact of multiple factors on an outcome variable.

8. Random Design - Merits & Demerits:

Merits:

• Reduces bias through random treatment assignment.
• Improves generalizability of results to the population.
• Simple to implement and statistically efficient (for few treatments).

Demerits:

• Limited control over extraneous variables.
• May not account for underlying group differences.
• Doesn't capture treatment interactions with other factors.
• Larger sample size might be needed compared to some designs.