What is p-value?

-A p-value is a statistical measure used in hypothesis testing

-Assess the strength of evidence against a null hypothesis.

-It represents the probability of obtaining results as extreme as, or more extreme than, the observed results under the assumption that the null hypothesis is true. -The smaller the p-value, the more likely you are to reject the null hypothesis.

Suppose two groups have test scores:

Group A: 88, 75, 90, 95, 80

Group B: 65, 70, 72, 78, 68

Are their means significantly different?

## 
##  Welch Two Sample t-test
## 
## data:  a and b
## t = 3.5735, df = 6.6043, p-value = 0.009982
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##   4.952376 25.047624
## sample estimates:
## mean of x mean of y 
##      85.6      70.6

Suppose a company claims that their batteries last at least 100 hours on average.
You test 30 batteries and get:

• Sample mean: \(\bar{x} = 98.5\) hours
• Known standard deviation: \(\sigma = 4\) hours

Is there evidence at \(\alpha = 0.05\) to doubt the company’s claim?

We use the one-sample z-test for the mean.

\[ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \]

• \(\bar{x} = 98.5\) (sample mean)
• \(\mu_0 = 100\) (claimed mean)
• \(\sigma = 4\) (known std dev)
• \(n = 30\) (sample size)

\[ z = \frac{98.5 - 100}{4/\sqrt{30}} \approx -2.06 \]

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
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## Warning: Removed 130 rows containing non-finite outside the scale range
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## Warning: Removed 130 rows containing non-finite outside the scale range
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Since we’re checking if the mean is less than claimed (one-sided test):

\[ p\text{-value} = P(\bar{X} \leq 98.5 \mid H_0) = P(Z \leq -2.06) \]

• The area in red is the p-value.

• Calculated p-value: \(p = 0.0197\) (from \(Z = -2.06\))
• Since \(p = 0.0197 < \alpha = 0.05\), reject \(H_0\)
• There is statistically significant evidence to doubt the company’s claim.