Answer to Question #137075 in Statistics and Probability for prasant kandel

Question

Q1. A low noise transistor for use in computing products is being developed. It is claimed that the mean
noise level will be below the 2.5 dB level of the product currently in use.
a) Set up the appropriate null and alternative hypotheses for verifying the claim
b) A sample of 16 transistors yields x̅= 1.8 and s = 0.8. Find the P value for the test. Do you think
that H0 should be rejected? What assumptions are making concerning the distribution of the
random variable X, the noise level of a transistor?
c) Explain the context of this problem, what conclusions can be drawn regarding the noise level of
these transistors. If you make a Type I error, what will have occurred? What is the probability
that you are making such as error?

Answer 1

The provided sample mean is $\bar{X}=1.8$ and the sample standard deviation is $s=0.8,$ and the sample size is $n=16.$

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

$H_0 : \mu\geq2.5$

$H_1:\mu<2.5$

This corresponds to a left-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used.

Based on the information provided, the significance level is $\alpha=0.05,$ and the critical value for a left-tailed test is $t_c=-1.753.$  

The rejection region for this left-tailed test is $R=\{t:t<-1.753\}.$  

The t-statistic is computed as follows:

$$t=\dfrac{\bar{X}-\mu}{s/\sqrt{n}}=\dfrac{1.8-2.5}{0.8/\sqrt{16}}=-3.5$$

Since it is observed that $t=-3.5<-1.753=t_c,$ it is then concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is less than 2.5, at the 0.05 significance level.

The p-value is $p=0.001612,$ and since $p=0.001812<0.05,$ it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean $\mu$ is less than 2.5, at the 0.05 significance level.

When the null hypothesis is true and you reject it, you make a type I error. The probability of making a type I error is $\alpha,$ which is the level of significance you set for your hypothesis test. An $\alpha$ of 0.05 indicates that you are willing to accept a 5% chance that you are wrong when you reject the null hypothesis. To lower this risk, you must use a lower value for $\alpha.$ However, using a lower value for alpha means that you will be less likely to detect a true difference if one really exists.