Hypothesis Testing

To test if there is a difference in the average response times of two call centers, you can use a two-sample t-test. The null hypothesis (H0) is typically that the means are equal, and the alternative hypothesis (Ha) is that the means are not equal (two-tailed test).

Here are the given values: For Center A:

• Sample size (n1) = 40
• Sample mean (xˉ1​) = 4 minutes
• Sample standard deviation (s1) = 1 minute

For Center B:

• Sample size (n2) = 35
• Sample mean (xˉ2​) = 4.5 minutes
• Sample standard deviation (s2) = 1.2 minutes

Significance level ($\alpha$) = 0.05 (5%)

To calculate the t-statistic for a two-sample t-test, use the following formula:

t=(x1−x2)/√x1^2/n1+x2^2/n2

$t\approx -2.3946$

The calculated t-statistic for this two-sample t-test is approximately -2.3946.

Please note that the t-statistic is negative because xˉ1​ is less than xˉ2​, indicating that the average response time for Center A is lower than that of Center B.

To conduct a hypothesis test to determine if the company’s claim that the product has a failure rate of less than 5% is supported, you can use a one-sample z-test. The null hypothesis (H0) is that the failure rate is 5% or higher, and the alternative hypothesis (Ha) is that the failure rate is less than 5%.

Here are the given values:

• Sample size (n) = 200
• Number of failures in the sample (x) = 15
• Population proportion under the null hypothesis (p0​) = 0.05 (5%)
• Significance level ($\alpha$) = 0.01 (1%)

To calculate the z-statistic, use the formula for a one-sample z-test:

z = $\frac{p-p_{\mathrm{0/}}}{\sqrt{\frac{p_0(1-p_0)/}{n}}}$

Here, p is the sample proportion of failures (x$/n$).

$p_0$ is the hypothesized population proportion (0.05).

$z\approx 2.68$

The calculated z-statistic for this test is approximately 2.68.

To conduct a hypothesis test to determine if the new engine design increases fuel efficiency, you can use a one-sample t-test. The null hypothesis (H0) is that the new design has no effect on fuel efficiency, and the average fuel efficiency is 30 miles per gallon (mpg). The alternative hypothesis (Ha) is that the average fuel efficiency with the new engine design is different from 30 mpg.

Here are the given values:

• Sample size (n) = 36
• Sample mean (xˉ) = 32 mpg
• Sample standard deviation (s) = 3 mpg
• Population mean ($\mu$) under the null hypothesis = 30 mpg
• Significance level ($\alpha$) = 0.01 (1%)

To calculate the t-statistic, use the formula:

t=(​xˉ−μ)/s/sqrt(n)​

Where:

• xˉ is the sample mean.
• $\mu$ is the hypothesized population mean (30 mpg).
• $s$ is the sample standard deviation.
• $n$ is the sample size.

Plug in the values:

t=4​

�=236t=63​2​

�=20.5t=0.52​

$t=4.0$

The calculated t-statistic for this test is 4.0.

In hypothesis testing, the p-value is a measure of the strength of evidence against the null hypothesis (H0). It tells you the probability of observing the data or something more extreme under the assumption that the null hypothesis is true. The significance level (α) is the threshold set by the researcher to determine whether to reject the null hypothesis or not.

In your case, the researcher has set a significance level (α) of 0.05, and the calculated p-value is 0.032. To make an appropriate conclusion:

1. If the p-value is less than or equal to the significance level (α), you reject the null hypothesis (H0).
2. If the p-value is greater than the significance level (α), you fail to reject the null hypothesis (H0).

In your case, the p-value (0.032) is less than the significance level (α = 0.05). Therefore, you should reject the null hypothesis.

So, the appropriate conclusion is to reject the null hypothesis at the 0.05 significance level.

In a one-sample z-test, you compare the calculated z-statistic to a critical z-value (z-critical) determined by the chosen significance level (α) to make a conclusion. In this case, the null hypothesis (H0) states that the population mean is 50, and the alternative hypothesis (Ha) suggests that the population mean is greater than 50. You have calculated a z-statistic of 2.5.

To determine the appropriate conclusion, you need to find the critical z-value for a one-tailed test at a 0.01 significance level. The critical z-value can be found using a standard normal distribution table or calculator.

For a one-tailed test with a 0.01 significance level, the critical z-value is approximately 2.33.

Now, compare the calculated z-statistic to the critical z-value:

• Calculated z-statistic = 2.5
• Critical z-value = 2.33

Since the calculated z-statistic (2.5) is greater than the critical z-value (2.33), you should reject the null hypothesis (H0) in favor of the alternative hypothesis (Ha).

So, the appropriate conclusion is to reject the null hypothesis at the 0.01 significance level.