Respuesta :
Step-by-step explanation:
When interpreting results in research, we need to determine whether if such findings may have occurred by chance. Hence, hypothesis testing is a methodical procedure for establishing if the results actually support a particular theory (hypothesis) in relation to a sample population.
Hypothesis testing first transforms the research question into two concise statements, usually in mathematical form, a null hypothesis, [tex]H_{0}[/tex], and an alternative hypothesis, [tex]H_{A}[/tex]. These statements confer the 2 possible versions of the "truth" between the prediction of interest and the actual outcome within the population to which the null hypothesis describes the lack of association between the prediction and the outcome, whereas the alternative hypothesis describes the existence of an association between the prediction and the outcome.
In this particular case, [tex]\mu[/tex] represents the average weight of cereal boxes at his local grocery store,
[tex]H_{0}: \ \mu \ = \ 25 \ \\ \-\hspace{0.9cm} \text{(There is \textbf{no significance difference} between the average weight of a} \\ \-\hspace{0.9cm} \text{cereal box and the weight of a cereal box sold at the local grocery store.)}\\\\ \\ \\H_{A}: \ \mu \ < \ 25 \\ \-\hspace{0.9cm} \text{(There is \textbf{significance difference} between the average weight of a} \\ \-\hspace{0.9cm} \text{cereal box and the weight of a cereal box sold at the local grocery store.)}[/tex]
To set the criteria for a decision, a level of significance (critical value) is stated for the test. Generally, the level of significance is set at 5% or 0.05 such that when the probability of obtaining the sample mean (not the mean of the actual outcome) is less than 0.05, the null hypothesis is then rejected as a decisive truth. On the other hand, the truth then follows the alternative hypothesis.
Additionally, we need to determine the degrees of freedom, df , which is given by the formula
[tex]\textit{\textbf{df}} \ = \ N \ - \ 1[/tex],
where [tex]N[/tex] is the total number of observations.
Therefore, to obtain the p-value of the t-test, we can refer to a one-tailed t-distribution table online, since the t-value is given as -2.624 which is located at the lower tail of the distribution. Considering the symmetry of the t-distribution around the mean, similar to the normal distribution, it is also equivalent to the t-value 2.624. According to the distribution table, for a t-value of 2.624 with df = 14, the corresponding critical value is 0.01 (hundredths place).
Since, the p-value of the test is less than 0.05 (0.01 < 0.05), we can reject the null hypothesis, and therefore we can conclude that, according to the results obtained, the cereal boxes have an average weight which is less than 25 ounces.
Alternatively, using Python, we can find a more accurate p-value of the corresponding t-value than in a typical t-table by using the scipy package according to the code shown in the attached image and the output.
