A leading company chemically treats its product before packaging. The company monitors the weight
of product per hour that each machine treats. An SRS of 90 hours of production data for a particular
machine is collected. The measured variable is in pounds.
(a) Using the data �le 'capacity.xls', create a histogram, a boxplot, and a Normal quantile
plot of these data. From inspecting the plots, comment on skewness, Normality, and the presence
of outliers. Is it appropriate to analyze these data using t distribution methods? Explain.
(b) Test whether these data provide evidence that the mean pounds of product treated in
one hour is greater than 33000.
Use a signi�cance level of 5% and state your hypotheses, the P-value, and your conclusion.
We do not have access to your data.
To answer these questions, the following steps can be taken:
(a) Data Visualization and Analysis:
1. Import the 'capacity.xls' file into your statistical software tool.
2. Extract the column containing the weight of product per hour data.
3. Create a histogram using the weight data to visualize the distribution. This will show the frequency and shape of the data.
4. Create a boxplot to look for outliers and to assess skewness. The boxplot will display the minimum and maximum values, quartiles, and any potential outliers.
5. Construct a Normal quantile plot (also known as a Normal probability plot or Q-Q plot) to assess the normality of the data distribution. This plot compares the observed data quantiles to the expected quantiles of a normal distribution.
Based on these plots:
- Skewness: The histogram will give you an indication of the data's skewness. If the data is symmetric and bell-shaped, it is likely normally distributed.
- Normality: The Normal quantile plot will show the deviation of the data from a straight line. If the points on the plot closely align with the diagonal line, the data is likely normally distributed.
- Outliers: The boxplot will help identify any extreme values that may be considered outliers.
Whether it is appropriate to use t-distribution methods or not depends on the shape of the distribution and the presence of outliers. If the data appears to be normally distributed and does not exhibit extreme outliers, using t-distribution methods would be appropriate. If the data is heavily skewed or contains outliers, alternative distributional assumptions or non-parametric tests may be considered.
(b) Hypothesis Testing:
1. Formulate the null (H0) and alternative (H1) hypotheses. In this case:
- H0: The mean pounds of product treated in one hour is not greater than 33000.
- H1: The mean pounds of product treated in one hour is greater than 33000.
2. Conduct a one-sample t-test to test the hypothesis using the collected data.
3. Set the significance level to 5% (0.05).
4. Calculate the test statistic and the corresponding p-value.
5. Compare the p-value with the significance level to determine whether to reject or fail to reject the null hypothesis.
- If the p-value is less than 0.05, reject the null hypothesis.
- If the p-value is greater than or equal to 0.05, fail to reject the null hypothesis.
6. State your conclusion based on the results of the hypothesis test.
By following these steps, you will be able to analyze the data and draw conclusions about skewness, normality, outliers, and conduct hypothesis testing to determine whether the mean pounds of product treated in one hour is greater than 33000.