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The Z-test is a statistical method used to determine if there is a significant difference between the means of two datasets, typically when the sample size is large. In this context, it's applied to check if the average time spent on the new web page is significantly different from the old one.
In this exercise, the Z-test is suitable because the sample size is quite large (7,628), and we know the sample standard deviation. The formula for the Z-test statistic is:

• \[ Z = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \]

Here, \( \bar{x} \) is the sample mean (23.5 seconds), \( \mu \) is the population mean (23.6 seconds), \( s \) is the standard deviation (5.1 seconds), and \( n \) is the sample size (7,628).
By plugging these numbers into the formula, we calculate the Z value to compare against a critical value, helping us decide whether the new webpage performs better or worse.

The p-value helps us determine the strength of the results in hypothesis testing. It indicates the probability of obtaining test results at least as extreme as the observed results, under the assumption that the null hypothesis is correct.
In simpler terms, a smaller p-value means that there is stronger evidence in favor of the alternative hypothesis. For this exercise, after calculating the Z value, we found a p-value of approximately 0.043. This was derived from the Z-table, based on the Z value of -1.715.

• A p-value less than the significance level (0.05) suggests rejecting the null hypothesis.
• This implies it's unlikely that such a mean difference would occur by chance, providing us a reason to believe the new web page's average visit time is indeed less.

Hence, the p-value supports our conclusion from the critical value approach in rejecting the null hypothesis.

The null hypothesis (\(H_0\)) is a statement that suggests no effect or no difference in context to what is being tested. It acts as a starting assumption for hypothesis testing.
For this problem, the null hypothesis was set as the average visit time for the new page being equal to 23.6 seconds, similar to the old page. It's crucial to have the null hypothesis because it provides a specific benchmark against which the actual data is tested.

• In hypothesis testing, if the evidence strongly contradicts the null hypothesis, we reject it.
• If not, we do not have enough evidence to reject it.

The alternative hypothesis (\(H_a\)), in this case, suggests the new page's mean time is less than the old time. Thus, hypothesis testing helps us confirm or refute this alternative scenario.

Standard deviation is a key statistic that indicates how much the individual data points differ from the average (mean). It's a way of quantifying the amount of variation or dispersion in a set of values.
In this exercise, the standard deviation of 5.1 seconds gives us an idea of how much the visit times vary among the visitors of the new web page. A higher standard deviation would imply more variation, whereas a lower one suggests that visit times are more consistent.

• When conducting a Z-test, standard deviation plays a vital part in deciding how spread out the data is in comparison to the population mean.
• It also influences the test statistic, as a smaller standard deviation can make even small differences in means appear significant, and vice versa.

Hence, understanding standard deviation in context helps interpret the significance of the results from the hypothesis testing scenarios.