Step 3: Calculating the population mean This step will calculate the mean for the population data. Click the block of code below and hit the Run button above. # You can use the "mean" method of a pandas dataframe. pop_mean tpcp_df[*TPCP'].mean() print("Population mean = round (pop_mean, 2)) Population mean = 499.57 Step 4: Drawing one random sample from the population data and calculating the sample mean This block of code randomly selects one sample (with replacement) of 50 data points from the population data. Then it calculates the sample mean. You will use the "sample" method of the dataframe to select the sample. Click the block of code below and hit the Run button above. #use sample method of the dataframe to select a random sample, with replacement, of size 58. tpcp_sample_df = tpcp_df.sample (50, replace=True) #print the sample mean. sample_nean = tpcp_sample_df['TPCP'].mean() print("Sample mean =, round(sample_mean,2)) Sample mean = 514.42 Step 5: Repeatedly drawing samples and saving the sample mean for each sample You will now essentially repeat Step 4 one thousand times to select 1,000 random samples, with replacement, of size 50 from the population data. The code below contains a loop so that you can do this selection with just one click! You will save the sample mean for each sample in a Python dataframe. Click the block of code below and hit the Run button above. #run a for Loop to repeat the process Step 4 one thousand times to select one thousand samples. # save the mean of each sample that was drawn in a Python List called means_list. means_list = [] for i in range (1000): tpcp_sample_df = tpcp_df.sample(50, replace=True) sample_mean = tpcp_sample_df[*TPCP'].mean() means_list.append(sample_mean) #create a Python dataframe of means. means_df = pd.DataFrane (means_list, columns=['means']) print("Dataframe of 1000 sample means\n") print(means_df) Dataframe of 1000 sample means means 0 556.54 1 536.96 2 501.16 3 560.72 4 486.76 995 472.00 996 472.00 997 471.42 998 533.34 999 568.28 [1000 rows x 1 columns] Step 6: Creating a histogram plot of the sample means from Step 5 Now you will plot the data distribution of the 1,000 means from Step 5. View the plot to confirm that it approximates a Normal distribution (bell-shaped curve). Note that the original data distribution in Step 2 was skewed. However, the distribution of sample means, calculated by repeatedly drawing large samples, is approximately Normally distributed. Click the block of code below and hit the Run button above. NOTE: If the graph is not created, click the code section and hit the Run button again. #create a figure for the plot. fig, ax = plt.subplots() # create a histogram plot with 50 bins of 1,000 means. plt.hist(means_df['means'], bins=58) #set a title for the plot, x-axis and y-axis. plt.title('Distribution of 1888 sample means', fontsize=20) # title ax.set_xlabel('Means') ax.set_ylabel('Frequency') #show the plot. plt.show() Distribution of 1000 sample means 350 400 450 500 550 600 650 Means Step 7: Mean and the standard deviation of the sample mean distribution Now you will calculate the "grand" mean ("grand" because it is the mean of the 1,000 means) and the standard deviation of 1,000 sample means. Note that the distribution of sample means was approximately Normal (bell-shaped) in Step 6. Therefore, calculating the mean and the standard deviation of this distribution will allow us to calculate probabilities and critical values. Click the block of code below and hit the Run button above. # calculate mean of the 1,000 sample means (this is called the grand mean or mean of the means). mean1000 = means_df ['means'].mean() print("Grand (Mean of 1000 sample means) =",round (mean1000,2)) #calculate standard deviation of the 1,000 sample means. std1000= means_df['means'].std() print("Std Deviation of 1000 sample means =",round(std1000,2)) #print the probability that a specific mean is 450 or Less for a Normal distribution with mean and standard deviation of 1,000 prob_450_less_or_equal = st.norm.cdf (450, mean1000, std1000) print("Probability that a specific mean is 458 or less round (prob_450_less_or_equal,4)) Grand Mean (Mean of 1000 sample means) = 498.37 Std Deviation of 1000 sample means = 52.41 Probability that a specific mean is 450 or less = 0.178