Question

It is believed that a stock price for a particular company will grow at a rate of $5$per week with a standard deviation of$1$. An investor believes the stock won’t grow as quickly. The changes in stock price are recorded for ten weeks and are as follows: $4$,$3$,$2$, $3$, $1$,$7$, $2$,$1$,$1$, $2$. Perform a hypothesis test using a $5\%$level of significance. State the null and alternative hypotheses, find the $p$-value, state your conclusion, and identify the Type $\text{I}$and Type $\text{II}$errors.

Short answer

Type I error is: The stock price of a particular company actually rises at a rate of $5$per week, but never at a rate of $5$per week.

Type II error is: The stock price of a particular company does not actually rise at a rate of $5$per week, but it does rise at a rate of $5$per week.

Step-by-step solution

Given Information

This is a check of a single population mean, due to the fact the hassle is set imply modifications inventory price.

We need to test

$H_0:\mu =5$versus $H_a:\mu <5$

using $\alpha =0.05$

Investors do not believe stock prices will grow that fast. Therefore, the$<$says that it is left.

Random variable $\overline{X}$is the mean stock price. Distribution for the test is normal because we know the standard deviation i.e.

$\overline{X}:\mathcal{N}\left(\mu ,\frac{\sigma }{\sqrt{n}}\right)=\mathcal{N}\left(5,\frac{1}{\sqrt{10}}\right)$

Now we calculate the p-value using the normal distribution for a mean:

$P$-value$=P(\overline{x}<2.6)\approx 0$

where the sample mean of the problem is given by

$\frac{2+3+\cdots +1+2}{10}=2.6$

The $P$-value denotes the probability to the left of the sample mean in the normal distribution.

Step 2:  Explanation

We can consider that,

$\alpha =0.05>0=\mathrm{P}$-value

Therefore, we reject $H_0:\mu =5$. In other words, we do not think a stock price for a particular company will grow at a rate of $5$per week but believes the stock won't grow as quickly.

At the $5\%$significance level, we conclude that is not sufficient evidence to conclude that a stock price for a particular company will grow at a rate of $5$per week.

Rejecting the null hypothesis $H_0$when it is true is defined as a type $\text{I}$error.

Suppose the null hypothesis ${\mathrm{H}}_0$is as follows:" The stock price of a particular company rises once a week. "

So, Type $\text{I}$error is -. Reject the null hypothesis that a stock price for a particular company will not grow at a rate of $5$per week when is actually will grow at a rate of $5$per week.

Failing to reject the null hypothesis when it is false is defined as a type $\text{II}$error.

Therefore, Type $\text{II}$error is

Not to reject the null hypothesis that a stock price for a particular company will grow at a rate of $5$per week when is actually will not grow at a rate of $5$per week.

Conclusion

The graph for this problem is: