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An experiment to compare the tension bond strength of polymer latex modified mortar (Portland cement mortar to which polymer latex emulsions have been added during mixing) to that of unmodified mortar resulted in \(\bar{x}=18.12 \mathrm{kgf} / \mathrm{cm}^{2}\) for the modified mortar ( \(m=40\) ) and \(\bar{y}=16.87 \mathrm{kgf} / \mathrm{cm}^{2}\) for the unmodified mortar ( \(n=32\) ). Let \(\mu_{1}\) and \(\mu_{2}\) be the true average tension bond strengths for the modified and unmodified mortars, respectively. Assume that the bond strength distributions are both normal. a. Assuming that \(\sigma_{1}=1.6\) and \(\sigma_{2}=1.4\), test \(H_{0}: \mu_{1}-\mu_{2}=0\) versus \(H_{2}: \mu_{1}-\mu_{2}>0\) at level .01. b. Compute the probability of a type II error for the test of part (a) when \(\mu_{1}-\mu_{2}=1\). c. Suppose the investigator decided to use a level \(.05\) test and wished \(\beta=, 10\) when \(\mu_{1}-\mu_{2}=1\). If \(m=40\), what value of \(n\) is necessary? d. How would the analysis and conclusion of part (a) change if \(\sigma_{1}\) and \(\sigma_{2}\) were unknown but \(s_{1}=1.6\) and \(s_{2}=1.4\) ?

Step by step solution

01

Define the Hypothesis and Set the Significance Level for Part (a)

In this step, we establish the null hypothesis \(H_0 : \mu_1 - \mu_2 = 0\) and the alternative hypothesis \(H_a : \mu_1 - \mu_2 > 0\). The significance level \(\alpha\) is given as 0.01.

02

Calculate the Test Statistic for Part (a)

The test statistic for the difference in means, assuming known standard deviations, is given by:\[Z = \frac{(\bar{x} - \bar{y}) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{m} + \frac{\sigma_2^2}{n}}}\]Substitute \(\bar{x} = 18.12\), \(\bar{y} = 16.87\), \(\sigma_1 = 1.6\), \(\sigma_2 = 1.4\), \(m = 40\), and \(n = 32\):\[Z = \frac{(18.12 - 16.87) - 0}{\sqrt{\frac{1.6^2}{40} + \frac{1.4^2}{32}}} = \frac{1.25}{0.362} \approx 3.45\]

03

Determine the Critical Value and Decision for Part (a)

The critical value for a one-tailed test at \(\alpha = 0.01\) is approximately \(Z_{0.01} = 2.33\). Since the calculated \(Z\)-value (3.45) is greater than 2.33, we reject the null hypothesis.

04

Calculate Type II Error Probability for Part (b)

To find \(\beta\), the probability of a Type II error when \(\mu_1 - \mu_2 = 1\), we use:\[\beta = P\left(Z < Z_{0.01} - \frac{1}{\sqrt{\frac{\sigma_1^2}{m} + \frac{\sigma_2^2}{n}}} \right)\]Substitute the values:\[\beta = P\left(Z < 2.33 - \frac{1}{0.362} \right) \approx P(Z < -0.437) \approx 0.3315 \]

05

Compute Sample Size for Desired Power in Part (c)

The investigator wants \(\beta = 0.10\) with \(\alpha = 0.05\). First calculate \(\text{power} = 1 - \beta = 0.90\). Using a standard normal distribution table, for \(\beta = 0.10\), \(Z_{\beta} = 1.28\), and for \(\alpha = 0.05\), \(Z_{\alpha} = 1.645\). Solve for \(n\):\[1 = (Z_{\alpha} + Z_{\beta}) \sqrt{\frac{1.6^2}{m} + \frac{1.4^2}{n}}\]Rearrange and solve the above equation, \[n = \left(\frac{1}{(1.645 + 1.28) \times 0.313}\right)^2 \approx 25.44\]Thus, \(n\) needs to be at least 26.

06

Address Unknown Variances for Part (d)

If \(\sigma_1\) and \(\sigma_2\) are unknown, we replace them by their sample counterparts \(s_1\) and \(s_2\). Use the t-distribution instead of the Z-distribution. Compute a pooled standard deviation and a t-statistic with \(m+n-2\) degrees of freedom. The calculation follows a similar form as the Z-test but references t-tables for critical values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Type I Error

In hypothesis testing, a Type I error occurs when we incorrectly reject a true null hypothesis. It is often referred to as a "false positive". Imagine conducting a test to determine if a new teaching method is more effective than the current one. If we conclude it is, when in fact it isn't, we've made a Type I error.

The probability of making a Type I error is denoted by \( \alpha \), the significance level of the test. For instance, setting \( \alpha = 0.01 \) implies we are willing to accept a 1% chance of incorrectly rejecting the null hypothesis.

• **Significance Level:** This is a predetermined threshold that dictates how much risk of a Type I error one is willing to accept.
• **Critical Value:** In a test, the critical value helps decide whether to reject the null hypothesis. If the test statistic exceeds this value, the null hypothesis is rejected.

Understanding and setting the right significance level is crucial, as it balances the risk of a Type I error against the need to detect real effects in experiments.

Type II Error

While Type I errors concern false positives, Type II errors focus on false negatives—failing to reject a false null hypothesis. In the mortar example, suppose the modified mortar really is stronger than the unmodified, but our test fails to show this difference. This situation results in a Type II error.

The probability of making a Type II error is denoted by \( \beta \). In practical terms, \( \beta \) represents the risk of missing an actual effect. For instance, suppose \( \beta = 0.3315 \), as calculated in the exercise. This means there is a 33.15% chance that the test will not detect a true difference when it exists.

• **Power of a Test:** Power is equal to \( 1-\beta \) and indicates the probability of correctly rejecting a false null hypothesis. Higher power is desired to ensure that tests are sensitive enough to detect actual differences.
• **Influencing Factors:** The sample size, significance level, and true effect size all impact \( \beta \).

Balancing Type I and Type II errors involves careful planning of experiments, aiming to minimize both effectively.

Sample Size Calculation

Determining the appropriate sample size for an experiment is essential in ensuring reliable and valid results. A well-calculated sample size helps balance the probabilities of Type I and Type II errors, whilst maintaining cost-effectiveness. In the given problem, the investigator sought a sample size \( n \) that would ensure the desired power of 0.90, translating to a Type II error probability \( \beta \) of 0.10.

To determine \( n \), factors such as the desired power, levels of significance \( \alpha \), and the effect size must be considered:

• **Effect Size:** This value represents the magnitude of the difference we are testing for. Larger effect sizes generally lead to smaller required sample sizes.
• **Significance Level and Power:** Increasing the significance level or the power of the test typically requires larger sample sizes.

In this scenario, the calculated necessary sample size was approximately 26 participants for the unmodified group, ensuring the study's effectiveness and reliability.

Pooled Standard Deviation

When dealing with hypothesis tests involving multiple samples, the pooled standard deviation is a vital concept, especially when the population standard deviations are assumed to be equal but unknown. It provides an estimate by combining the standard deviations from both groups. In situations of unequal sample sizes, pooled standard deviation offers an unbiased estimate of variability.

In the exercise, if \( \sigma_1 \) and \( \sigma_2 \) were unknown, the solution would have involved calculating a pooled standard deviation using sample standard deviations \( s_1 \) and \( s_2 \). This approach rests on:

• **Formula:** The formula for calculating the pooled standard deviation involves the degrees of freedom from each group and their respective variances.
• **Assumptions:** We assume all samples come from populations with equal variances.

Utilizing the pooled standard deviation is crucial for valid inference in many practical testing scenarios, ensuring that variability is accurately accounted for across groups.