Lab 3: One Sample Z and T tests NOTE: SPSS outputs are necessary to show full completion of the lab. Please paste all SPSS outputs into your lab report and submit the completed reports including all requested tables and graphs. deducted for each SPSS requested output that is not included in the submitted lab document. Dataset: This lab uses the dataset (HeartFailure) Datasets submodule. Instructions for opening the dataset in SPSS are found as follows. SPSS installed on a computer: Reference page 4 of the SPSS Instruction Manual SPSS running remotely: Reference the slide "Opening your Dataset Remotely in SPSS via Go Remote" in the document "SPSS using Citrix access guidelines" on Brightspace. Two hundred fifty heart failure patients participated in a study examining the associations between life habits, and physiological factors. Participants completed an online survey about life habits that included the Outdoor time, the amount of water drink and the level of happiness level etc. Additional data were collected for each subject's gender, age, height, and weight. 1. Let's explore the data and collect some information for the mean Water_drink. a) (1 point) For the variable Water_drink, identify the following values: (i) the sample size, (ii) the sample mean, (iii) the sample standard deviation. Round the sample mean and sample standard deviation to 2 decimal places. Provide the table where you found this information in SPSS on your output. b) (1 point) Make a Normal quantile plot of the data. Comment on any important features and on the distribution of the data. Attach your SPSS graph. 2. The researchers are concerned that the mean Water_drink examined in the study is too different from what has been found in other studies. They expected to see an average Water_drink of 2.05 liter. Based on our data, is there evidence that the true mean Water_drink for the population is significantly different than the expected average of 2.05 liter at a 5% significance level? The population standard deviation is 0.6 liter.a) Perform a hypothesis test BY HAND, completing all the following steps. i. (2 point) State your hypotheses. ii. (2 points) Calculate the test statistic BY HAND. Round to 2 decimal places and show your work. iii. (2 points) Write down the probability statement and calculate the p-value. Round to 4 decimal places and show your work. b) (3 points) Draw your conclusions from the test. c) (1 point) Based on part b), decide whether the researchers' concerns are warranted. In other words, given your results above, and that the researchers were expecting the mean Water_drink to be 2.05 liter, should they be worried? 3. We are still interested in testing the similar hypothesis as in problem 2, but we really do not have an accurate knowledge of the population standard deviation. a) (1 point) Based on the information given, answer the following: (i) How does this problem differ from Problem 2? (ii) Based on this difference, will the test statistic be different? State the name of the test statistic. b) Based on the data, conduct a hypothesis test (with a 5% significance level) to see if there is evidence that the population mean Water_drink is greater than 1.93 liter. USE SPSS to find the test statistic and P-value, and as with all work done in SPSS, include your output in your Brightspace submission. i. (1 point) State your null and alternative hypotheses. Be sure to use appropriate notation. ii. (2 points) Give the value of the test statistic (USE SPSS, you should NOT calculate this by hand). iii. (2 points) Write down the probability statement and the p-value (USE SPSS for the p-value). Round your answer to 4 decimal places and show your work.iv. (1 point) State your conclusion in terms of the story. Make sure to specify whether your conclusion refers to the population or the sample. c) (2 points) What is the 95% confidence interval for the population mean Water_drink? Keep in mind we still don't know the true population standard deviation. You should use the SPSS output from (b) and if you are confused, refer to page 11 in the SPSS Instruction Manual. Round your answer to 2 decimal places and show your work. 4. (2 points) Consider the restrictions for using the t test procedures (robustness guidelines), for this sample size. a) What do we need our data to look like for the t procedures to be appropriate? b) Is it appropriate to use the t procedures for this data? (Consider using a plot to determine if it is appropriate