### 4.23 Nutrition labels.

The nutrition label on a bag of potato chips says that a one ounce (28 gram) serving of potato chips has 130 calories and contains ten grams of fat, with three grams of saturated fat. A random sample of 35 bags yielded a sample mean of 134 calories with a standard deviation of 17 calories. Is there evidence that the nutrition label does not provide an accurate measure of calories in the bags of potato chips? We have verified the independence, sample size, and skew conditions are satisfied.

#### Goal

We want to determine if there is evidence that the nutrition label on potato chip bags does not provide an accurate measure of calories. To do this, we perform a hypothesis test.

#### Hypothesis

Null Hypothesis: The mean amount of calories in a potato chip bag is equal to the nutrition label value of 130

Alternative Hypothesis: The mean amount of calories in a potato chip bag is not equal to the nutrition label value of 130

$$H_0: \mu = 130$$
$$H_A: \mu \neq 130$$

#### Significance Level

$$\alpha = 0.05$$

#### Z-Score

$$Z = \frac{\bar{x} - x}{SE_\bar{x}}$$

pop_mean = 130
samp_mean = 134
SE = 17
samp_size = 35
zscore <- round((samp_mean - pop_mean)/(SE/sqrt(samp_size)),digits=2)
print(paste0("The z-score is ", zscore))
## [1] "The z-score is 1.39"

#### Probability Value

pvalue <- round(2*pnorm(-abs(zscore)), digits = 4)
print(paste0("The p-value is ", pvalue))
## [1] "The p-value is 0.1645"

#### Decision

Since $$p = 0.1645 > \alpha = 0.05$$, we fail to reject the null hypothesis. There is not enough evidence to suggest that the nutrition label on potato chip bags does not provide an accurate measure of calories.