Question

in an article appearing in todays health a writer states that the average number of calories in a serving of popcorn is 75. to determine if the average number of calories in a serving of popcorn is different from 75, a nutritionist selected a random sample of 20 servings of popcorn and computed the sample mean number of calories per serving to be 78 with a sample standard deviation of 7.

Explanation:

1. Set up the hypotheses:

• The null hypothesis \(H_0:\mu = 75\) (where \(\mu\) is the population - mean number of calories in a serving of popcorn).
• The alternative hypothesis \(H_1:\mu

eq75\).

1. Identify the test - statistic formula:

• Since the population standard deviation \(\sigma\) is unknown and we have a sample standard deviation \(s\), we use the t - test statistic. The formula for the t - test statistic is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(\mu\) is the hypothesized population mean, \(s\) is the sample standard deviation, and \(n\) is the sample size.

1. Substitute the given values into the formula:

• We are given that \(\bar{x} = 78\), \(\mu = 75\), \(s = 7\), and \(n = 20\).
• First, calculate the denominator \(s/\sqrt{n}=\frac{7}{\sqrt{20}}\approx\frac{7}{4.472}\approx1.565\).
• Then, calculate the t - statistic: \(t=\frac{78 - 75}{1.565}=\frac{3}{1.565}\approx1.929\).

Step1: Set up hypotheses

Null hypothesis \(H_0:\mu = 75\), alternative \(H_1:\mu
eq75\).

Step2: Select test - statistic

Use \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\) as \(\sigma\) is unknown.

Step3: Calculate denominator

\(s/\sqrt{n}=\frac{7}{\sqrt{20}}\approx1.565\).

Step4: Calculate t - statistic

\(t=\frac{78 - 75}{1.565}\approx1.929\).

Answer:

The t - statistic is approximately \(1.929\).