Answer to Question #271507 in Statistics and Probability for Dawnnn

Question

Prices of basic commodities are often greatly affected by typhoons. During the aftermath of recent typhoon, the prices of bangus were reported to have been sold at P20 - P25 more

Answer 1

Prices of basic commodities are often greatly affected by typhoons. During the aftermath of recent typhoon, the prices of bangus were reported to have been sold at P20 - P25 more per kilogram at local wet markets. However, the price was back to its normal mean price at P100 per kilo one week after. If a random sample of 10 bangus sales transactions from local wet markets have prices (in pesos per kilo) of 107. 115, 130, 95, 100, 105, 98, 116, 104, and 100. Assuming that the bangus prices are normally distributed, is there sufficient evidence to say that the mean price of bangus in these markets is greater than P100? Use a 0.05 level of significance. State the null and alternative hypotheses

We need to compute the sample mean and the sample standard deviation. The following table is needed to compute the sample and variance:

The sample mean is computed as:

$\bar{X} = \frac{1}{n} \sum^n_{i=1} X_i \\ = \frac{1076}{10} \\ = 107.6$

The sample variance s² is

$s^2 = \frac{1}{n-1}(\sum^n_{i=1} X^2_i - \frac{1}{n}(\sum^n_{i=1} X_i)^2) \\ = \frac{1}{10-1}(116736 -\frac{1076^2}{10}) \\ = 106.4889$

The sample standard deviation s is:

$s = \sqrt{s^2} = \sqrt{106.4889} = 10.3193 \\ H_0: \mu = 100 \\ H_1: \mu > 100$

The population standard deviation is unknown and n<30 hence t-distribution is applicable for hypothesis testing. So, a degree of freedom is used which is calculated as df = n-1= 10-1 =9.

α=0.05

The critical value for a right-tailed test is t_c= 1.833

Test-statistic:

$t = \frac{\bar{X} - \mu}{s / \sqrt{n}} \\ = \frac{107.6-100}{10.3193 / \sqrt{10}} \\ = 2.329$

Since it is observed that t = 2.329 > t_c = 1.833, it is then concluded that the null hypothesis is rejected.

So, that the mean price of bangus in these markets is greater than P100 at 0.05 level of significance.