Q: Given the sample strengths for Cojet: 15.4, 12.9, 17.2, 16.6, 19.3, which of the following is the correct null and alternative hypothesis for testing if the mean shear bond strength differs from 15 MPa?

$$H_0$$: \mu = 15$$ vs H_a$$: \mu \neq 15$$

Question 1

A company claims the average shear bond strength of repaired veneers is 15 MPa. A pilot sample of 5 specimens repaired with the Cojet kit produced strengths: 15.4, 12.9, 17.2, 16.6, 19.3. What is the correct formula for the 95% confidence interval for the mean shear bond strength $$\mu of $$Cojet, assuming the population is normal and the sample standard deviation is $$s$$?

Accepted Answer

$$\left( \bar{x} \pm t_{0.025, n-1} \frac{s}{\sqrt{n}} \right)$$

Question 2

Given the sample strengths for Cojet: 15.4, 12.9, 17.2, 16.6, 19.3, which of the following is the correct null and alternative hypothesis for testing if the mean shear bond strength differs from 15 MPa?

Accepted Answer

$$H_0$$: \mu = 15$$ vs H_a$$: \mu 
eq 15$$

Question 3

To construct a 95% confidence interval for the variance $$\$$sigma^2$$ of Cojet’s shear bond strength, which formula should be used?

Accepted Answer

$$\left( \frac{(n-1)s^2$}{\$$chi^2$$_{0.025, n-1}}, \frac{(n-1)s^2$}{\$$chi^2$$_{0.975, n-1}} \right)$$

Question 4

Suppose the company requires that the variance in shear bond strength not exceed $$\$$sigma^2$$_0 = 2.5 ($$MPa$$^2). $$Which test statistic is used to test $$H_0$$: \$$sigma^2 = 2.5$ vs H_a$$: \$$sigma^2$$ 
eq 2.5$$?

Accepted Answer

$$\$$chi^2$$ = \frac{(n-1)s^2$}{\$$sigma^2$$_0}$$

Question 5

Researchers compare Cojet vs. Silistor using a two-sample t-test for equal means. Which formula correctly computes the pooled variance $$s_p^2$?

Accepted Answer

$$s_p^2$ = \frac{(n_1$ - 1)s_1$^2 + (n_2$ - 1)s_2$^2}{n_1$ + n_2$ - 2}$$

Question 6

In a two-sample t-test for equal means, what is the correct formula for the test statistic?

Accepted Answer

$$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1$} + \frac{1}{n_2$}}}$$

Question 7

To construct a 95% confidence interval for $$\mu_{Cojet} - \mu_{Silistor}$$, which formula should be used?

Accepted Answer

$$\left( (\bar{x}_1 - \bar{x}_2) \pm t_{0.025, df} \cdot s_p \sqrt{\frac{1}{n_1$} + \frac{1}{n_2$}} \right)$$

Question 8

For testing equality of variances between Cojet and Silistor, which formula is used for the F-test statistic?

Accepted Answer

$$F = \frac{s_1$^2}{s_2$^2}$$

Question 9

When constructing a 95% confidence interval for the ratio of population variances $$\frac{\$$sigma_1$$^2}{\$$sigma_2$$^2}$$, which formula is correct?

Accepted Answer

$$\left( \frac{s_1$^2}{s_2$^2} \cdot \frac{1}{F_{0.975, n_1$-1, n_2$-1}}, \frac{s_1$^2}{s_2$^2} \cdot \frac{1}{F_{0.025, n_1$-1, n_2$-1}} \right)$$

Question 10

In a one-way ANOVA comparing four repair kits, what is the correct formula for the F statistic?

Accepted Answer

$$F = \frac{MS_{Treatment}}{MS_{Error}}$$

Question 11

In the ANOVA table, how is the mean square for treatment ($$MS_{Treatment}) $$calculated?

Accepted Answer

$$MS_{Treatment} = \frac{SS_{Treatment}}{df_{Treatment}}$$

Question 12

In a one-way ANOVA with four kits and five samples each, what are the correct degrees of freedom for treatment and error?

Accepted Answer

Treatment: $$df = 3$$, Error: $$df = 16$$

Question 13

Which of the following best describes the experimental units, response variable, and factor in the one-way ANOVA for the four repair kits?

Accepted Answer

Experimental units: specimens; Response variable: shear bond strength; Factor: repair kit

Question 14

If the p-value in the ANOVA table is less than 0.05, what is the correct decision regarding the global hypothesis $$H_0$$: \mu_{Cojet} = \mu_{Silistor} = \mu_{Cimara} = \mu_{Ceramic\ Repair}$$?

Accepted Answer

Reject $$H_0$$; at least one kit's mean shear bond strength differs.

Question 15

In the context of the ANOVA, what is the correct formula for the total sum of squares ($$SS_{Total}$$)?

Accepted Answer

$$SS_{Total} = SS_{Treatment} + SS_{Error}$$

Question 16

A researcher wants to test if the mean shear bond strength of Cojet is significantly different from 15 MPa. Which critical value should be used for a two-sided t-test at $$\alpha = 0.05$$ with 4 degrees of freedom?

Accepted Answer

$$t_{0.025, 4}$$

Question 17

In a two-sample F-test for equality of variances, what are the numerator and denominator degrees of freedom?

Accepted Answer

Numerator: $$n_1 - 1$, Denominator: n_2$ - 1$$

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