Question

Football Air Pressure During the National Football League's 2014 AFC championship game, officials measured the air pressure on 11 of the game footballs being used by the New England Patriots. They found that the balls had an average air pressure of 11.1 psi, with a standard deviation of 0.40 psi. (a) Assuming this is a representative sample of all footballs used by the Patriots in the 2014 season, perform the appropriate test to determine if the average air pressure in footballs used by the Patriots was significantly less than the allowable limit of 12.5 psi. There is no extreme skewness or outliers in the data, so it is appropriate to use the \(\mathrm{t}\) -distribution. (b) Is it fair to assume that this sample is representative of all footballs used by the Patriots during the 2014 season?

Short answer

The t-test performed provided a t-value of -14.715 and a p-value that is significantly less than 0.05. This means the null hypothesis, which stated the average pressure of the footballs was not significantly less than 12.5 psi, can be rejected. However, it's not necessarily fair to consider the sample to be representative of all footballs used by the Patriots during the whole season due to potential variables affecting the footballs' pressure.

Step-by-step solution

- Identifying population mean, sample mean and sample standard deviation

We begin by identifying the value to test against, which is the population mean, which is 12.5 psi. Meanwhile, the sample mean (x̄), or the average pressure of the footballs in the sample, is 11.1 psi, and the standard deviation (s) is 0.40 psi.

- Select the Appropriate Test Statistic

Since we don't know the population variance and are dealing with a single sample, we shall use One Sample T-Test.

- Calculate the Test Statistic Value

We can calculate the t-value using the formula for one-sample t-test: \(t = \frac{{x̄ - μ}}{{s/√n}}\), where \(x̄\) is the sample mean, \(μ\) is the population mean, \(s\) is the standard deviation of the sample and \(n\) is the size of the sample. Here, \(n = 11\) (number of footballs), \(x̄ = 11.1\) psi (average pressure), \(s = 0.4\) psi (standard deviation) and \(μ = 12.5\) psi (population mean or allowable limit). Plugging in these values: \(t = \frac{{11.1 - 12.5}}{{0.4/√11}} = -14.715\).

- Determine the P-value

Next, we determine the p value. This cannot be calculated by hand and will depend on degree of freedom. In case of one sample t-test, the degrees of freedom is given by \(n - 1 = 11 - 1 = 10\). For a one-sided t-test with degree of freedom 10 and \(t = -14.715\), p < 0.0001. This p-value is significantly less than the commonly accepted threshold (0.05), thus, we would reject the null hypothesis of the average pressure not being less than the limit.

- Discussion on the representative nature of the sample

A sample is representative if it exhibits the characteristics of the population, allowing for accurate conclusions about the population based on the sample. In this case, it is not necessarily fair to assume that these 11 footballs accurately represent all footballs used by the Patriots in the 2014 season. Factors like the temperature at the time of the measurement, the length of time the balls were used in the game, and the general wear and tear on different balls may vary and affect their pressure. Therefore, while such assumptions may serve statistical tests, they should be made cautiously in realistic scenarios.