Melissa and Madeline love pepperoni pizza, but sometimes they are disappointed with the small number of pepperonis on their pizza. To investigate, they went to their favorite pizza restaurant at 10 random times during the week and ordered a large pepperoni pizza. Here are the numbers of pepperoni on each pizza: 47, 36, 25, 37, 46, 36, 49, 32, 32, 34. Using the step method, we will construct and interpret a 95% confidence interval for the true mean number of pepperonis on a large pizza at this restaurant.
The Correct Answer and Explanation is:
To construct a 95% confidence interval for the true mean number of pepperonis on a large pizza at the restaurant, we need to follow the steps of statistical inference. Here’s how you can do it using the step method:
Step 1: Identify the sample data
The data from the 10 random pizzas are as follows:
47, 36, 25, 37, 46, 36, 49, 32, 32, 34
Step 2: Calculate the sample mean (x̄)
First, we need to find the mean (average) of the sample:Sample Mean (xˉ)=∑(all data values)Number of observations=47+36+25+37+46+36+49+32+32+3410\text{Sample Mean (x̄)} = \frac{\sum \text{(all data values)}}{\text{Number of observations}} = \frac{47 + 36 + 25 + 37 + 46 + 36 + 49 + 32 + 32 + 34}{10}Sample Mean (xˉ)=Number of observations∑(all data values)=1047+36+25+37+46+36+49+32+32+34Sample Mean (xˉ)=40410=40.4\text{Sample Mean (x̄)} = \frac{404}{10} = 40.4Sample Mean (xˉ)=10404=40.4
So, the sample mean is 40.4 pepperonis.
Step 3: Calculate the sample standard deviation (s)
Next, we calculate the sample standard deviation. The formula for the sample standard deviation is:s=∑(xi−xˉ)2n−1s = \sqrt{\frac{\sum (x_i – \bar{x})^2}{n – 1}}s=n−1∑(xi−xˉ)2
Where:
- xix_ixi is each individual data point,
- xˉ\bar{x}xˉ is the sample mean,
- nnn is the number of observations (which is 10).
Now, let’s calculate the squared differences from the mean for each data point:
- (47 – 40.4)² = 43.56
- (36 – 40.4)² = 19.36
- (25 – 40.4)² = 240.16
- (37 – 40.4)² = 11.56
- (46 – 40.4)² = 30.56
- (36 – 40.4)² = 19.36
- (49 – 40.4)² = 73.96
- (32 – 40.4)² = 70.56
- (32 – 40.4)² = 70.56
- (34 – 40.4)² = 41.16
Now, sum up the squared differences:∑(xi−xˉ)2=43.56+19.36+240.16+11.56+30.56+19.36+73.96+70.56+70.56+41.16=620.4\sum (x_i – \bar{x})^2 = 43.56 + 19.36 + 240.16 + 11.56 + 30.56 + 19.36 + 73.96 + 70.56 + 70.56 + 41.16 = 620.4∑(xi−xˉ)2=43.56+19.36+240.16+11.56+30.56+19.36+73.96+70.56+70.56+41.16=620.4
Now, divide by n−1=9n – 1 = 9n−1=9 (since we have 10 data points):s=620.49=68.93≈8.3s = \sqrt{\frac{620.4}{9}} = \sqrt{68.93} \approx 8.3s=9620.4=68.93≈8.3
So, the sample standard deviation is 8.3.
Step 4: Find the critical value (t*)
Since we are constructing a 95% confidence interval with a sample size of 10, we use the t-distribution with 9 degrees of freedom (df = n – 1 = 10 – 1 = 9). For a 95% confidence interval, the critical value t* for 9 degrees of freedom can be found in a t-table or using a calculator. The value is approximately 2.262.
Step 5: Calculate the margin of error (ME)
The margin of error is calculated as:ME=t∗×snME = t^* \times \frac{s}{\sqrt{n}}ME=t∗×ns
Where:
- t∗=2.262t^* = 2.262t∗=2.262,
- s=8.3s = 8.3s=8.3,
- n=10n = 10n=10.
Now, plug the values in:ME=2.262×8.310=2.262×2.627≈5.94ME = 2.262 \times \frac{8.3}{\sqrt{10}} = 2.262 \times 2.627 \approx 5.94ME=2.262×108.3=2.262×2.627≈5.94
Step 6: Construct the confidence interval
Finally, the 95% confidence interval is:CI=xˉ±ME=40.4±5.94\text{CI} = \bar{x} \pm ME = 40.4 \pm 5.94CI=xˉ±ME=40.4±5.94
So, the confidence interval is:[40.4−5.94,40.4+5.94]=[34.46,46.34][40.4 – 5.94, 40.4 + 5.94] = [34.46, 46.34][40.4−5.94,40.4+5.94]=[34.46,46.34]
Step 7: Interpret the result
We are 95% confident that the true mean number of pepperonis on a large pizza at this restaurant lies between 34.46 and 46.34 pepperonis.
This means that if we were to repeat this process many times, approximately 95% of the calculated confidence intervals would contain the true average number of pepperonis on a large pizza at this restaurant.
