One-Tailed — Unknown Standard Deviation: Hypothesis Testing

In cases where the population standard deviation is not known, t-value is used as the test statistic for one-tailed tests. (Refer section t-test for details on the test and the t-value).

Example: To assess whether the proportions of surfactant in the detergent packs are correct, a study was conducted by technicians from a contracted lab. The study involved examining samples from 80 packs. According to the specifications, the packs should contain an average of 15% surfactant.

The findings from the sample indicate that the average surfactant composition in the sampled packs is 14.7 grams per 100 grams of detergent, with a standard deviation of 1.2 grams. Based on these findings, can we infer that the packs contain less than the specified level of surfactant?

H0: μ ≥ 15 gm per 100 gm of detergent

HA: μ < 15 gm per 100 gm of detergent

α = 5%

$$ t = \frac{\bar x-μ_0}{σ/\sqrt n} = \frac{14.7-15}{1.2/8.94} = -2.24 $$

Degrees of freedom = 79.

p-value = 0.014.

The probability of obtaining a t-value of −2.24 or lower, when sampling 80 packs from the population, has been determined to be 0.014 or 1.4%. This probability is lower than the significance level α of 5%. In other words, if the average surfactant concentration is assumed to be 15%, the probability that our sample would have an average of 14.7% or less is only 0.014 (1.4%). Based on this analysis, the null hypothesis is rejected, indicating that the data suggests that the manufacturer is not meeting the specified concentration of 15%.

Note: A p-value from t-value calculator is provided on this webpage.

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