A fast-food chain advertises that its large bag of french...

A fast-food chain advertises that its large bag of french fries has a weight of 150 grams. Some high school students, who enjoy french fries at every lunch, suspect that they are getting less than the advertised amount. With a scale borrowed from their physics teacher, they weigh a random sample of 15 bags. What is the conclusion if the sample mean is $145.8 \mathrm{~g}$ and standard deviation is $12.81 \mathrm{~g}$ ? (Assume that all conditions for inference are met.) (A) There is sufficient evidence to prove the fast-food chain advertisement is true. (B) There is sufficient evidence to prove the fast-food chain advertisement is false. (C) The students have sufficient evidence to reject the fastfood chain's claim. (D) The students do not have sufficient evidence to reject the fast-food chain's claim. (E) There is not sufficient data to reach any conclusion.

A fast-food chain advertises that its large bag of french fries has a weight of 150 grams. Some high school students, who enjoy french fries at every lunch, suspect that they are
getting less than the advertised amount. With a scale borrowed from their physics teacher, they weigh a random sample of 15 bags. What is the conclusion if the sample mean
is $145.8 \mathrm{~g}$ and standard deviation is $12.81 \mathrm{~g}$ ? (Assume that all conditions for inference are met.)
(A) There is sufficient evidence to prove the fast-food chain advertisement is true.
(B) There is sufficient evidence to prove the fast-food chain advertisement is false.
(C) The students have sufficient evidence to reject the fastfood chain's claim.
(D) The students do not have sufficient evidence to reject the fast-food chain's claim.
(E) There is not sufficient data to reach any conclusion.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or decisions about a population parameter based on sample data. It involves establishing a null hypothesis (a default position that there is no effect or no difference) and an alternative hypothesis (what the researcher aims to support), calculating a test statistic, and determining whether the observed data fall into a rejection region based on a pre-determined significance level.

Null and Alternative Hypotheses

The null hypothesis is the statement that there is no deviation from a claimed value, while the alternative hypothesis represents the claim of interest that challenges the status quo. In the context of the problem, the null hypothesis would state that the average weight is as advertised, and the alternative would contend that it is lower, setting the stage for a one-tailed test.

One-Sample t-Test

A one-sample t-test is used when comparing the sample mean to a known or claimed population mean, particularly when the population standard deviation is unknown and the sample size is small. It employs the t-distribution to account for extra variability, ensuring accurate inference when conditions for normality or when sample sizes are less than 30 are met.

Test Statistic and p-value

The test statistic, calculated from the sample data, quantifies how far the sample mean deviates from the claimed population mean in units of standard error. The p-value measures the probability of obtaining a test statistic at least as extreme as the observed one, assuming the null hypothesis is true. A low p-value indicates that the observed data is unlikely under the null hypothesis, leading to its rejection.

Degrees of Freedom

Degrees of freedom, typically calculated as the sample size minus one, reflect the number of independent values that can vary in the calculation of a statistic. In a t-test, they determine the shape of the t-distribution used for inference, affecting the critical values and ultimately the conclusions drawn from the statistical test.

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