Question

(1 point) (Hypothetical.) A certain type of soda has 200 calories in a one liter bottle, according to the label. A small simple random sample of such bottles has the following calorie measurements: 202 \ \ 205 \ \ 201 \ \ 202 \ \ 198 Four out of five measurements are over 200. Test the null hypothesis that the correct measurement is 200 calories and the different observations are due simply to chance measurement error. (Assume measurement error to be Normally distributed). Round your values to three decimal places.

Answer 1

Answer:

$p_v =2*P(t_{4}>1.425)=0.227$    

If we compare the p value and a significance level for example $\alpha=0.05$ we see that $p_v>\alpha$ so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, so we can conclude that the population mean is NOT significant different from 200 at 5% of significance.    

Step-by-step explanation:

Previous concepts  and data given  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

$\bar X=201.6$ represent the sample mean  

$s=2.510$ represent the sample standard deviation  

n=5 represent the sample selected  

$\alpha$ significance level  

State the null and alternative hypotheses.    

We need to conduct a hypothesis in order to check if we have significant difference on the mean of 200, the system of hypothesis would be:    

Null hypothesis:$\mu = 200$    

Alternative hypothesis:$\mu \neq 200$    

If we analyze the size for the sample is < 30 and we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:    

$t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}$  (1)    

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".    

Calculate the statistic  

We can replace in formula (1) the info given like this:    

$t=\frac{201.6-200}{\frac{2.510}{\sqrt{5}}}=1.425$      

P-value  

First we need to calculate the degrees of freedom given by:

$df=n-1=5-1 = 4$

Then since is a two sided test the p value would be:    

$p_v =2*P(t_{4}>1.425)=0.227$    

Conclusion  

If we compare the p value and a significance level for example $\alpha=0.05$ we see that $p_v>\alpha$ so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, so we can conclude that the population mean is NOT significant different from 200 at 5% of significance.