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Len [333]
3 years ago
7

Consider the following hypothesis test: H 0: 20 H a: < 20 A sample of 60 provided a sample mean of 19.5. The population stand

ard deviation is 1.8. a. Compute the value of the test statistic (to 2 decimals). If your answer is negative, use minus "-" sign. b. What is the p-value (to 3 decimals)? c. Using = .05, can it be concluded that the population mean is less than 20? d. Using = .05, what is the critical value for the test statistic (to 3 decimals)? If your answer is negative, use minus "-" sign.
Mathematics
1 answer:
viva [34]3 years ago
5 0

Answer:

We reject the null hypothesis and accept the alternate hypothesis. Thus, it be concluded that the population mean is less than 20.          

Step-by-step explanation:

We are given the following in the question:

Population mean, μ = 60

Sample mean, \bar{x} = 19.5

Sample size, n = 60

Alpha, α = 0.05

Population standard deviation, σ = 1.8

First, we design the null and the alternate hypothesis

H_{0}: \mu = 20\\H_A: \mu < 20

We use One-tailed z test to perform this hypothesis.

Formula:

z_{stat} = \displaystyle\frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} }

Putting all the values, we have

z_{stat} = \displaystyle\frac{19.5 - 20}{\frac{1.8}{\sqrt{60}} } = -2.151

Now, z_{critical} \text{ at 0.05 level of significance } = -1.64

Since,  

z_{stat} < z_{critical}

We reject the null hypothesis and accept the alternate hypothesis. Thus, it be concluded that the population mean is less than 20.

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Answer:

t=\frac{4.2-3.6}{\frac{1.4}{\sqrt{16}}}=1.714

Reject the null hypothesis if the observed "t" value is less than -2.131 or higher than 2.131  

Rejection Zone: t_{calculated} or t_{calculated}>2.131

In our case since our calculated value is not on the rejection zone we don't have enough evidence to reject the null hypothesis at 5% of significance.

Step-by-step explanation:

Previous concepts  and data given  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

\bar X=4.2 represent the sample mean  

s=1.4 represent the sample standard deviation  

n=16 represent the sample selected  

\alpha=0.05 significance level  

State the null and alternative hypotheses.    

We need to conduct a hypothesis in order to check if the mean is different from 3.6, the system of hypothesis would be:    

Null hypothesis:\mu = 3.6    

Alternative hypothesis:\mu \neq 3.6    

If we analyze the size for the sample is < 30 and we know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:    

t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}  (1)    

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".    

Calculate the statistic  

We can replace in formula (1) the info given like this:    

t=\frac{4.2-3.6}{\frac{1.4}{\sqrt{16}}}=1.714

Critical values

On this case since we have a bilateral test we need to critical values. We need to use the t distribution with df=n-1=16-1=15 degrees of freedom. The value for \alpha=0.05 and \alpha/2=0.025 so we need to find on the t distribution with 15 degrees of freedom two values that accumulates 0.025 of the ara on each tail. We can use the following excel codes:

"=T.INV(0.025,15)" "=T.INV(1-0.025,15)"

And we got t_{crit}=\pm 2.131    

So the decision on this case would be:

Reject the null hypothesis if the observed "t" value is less than -2.131 or higher than 2.131  

Rejection Zone: t_{calculated} or t_{calculated}>2.131

Conclusion    

In our case since our calculated value is not on the rejection zone we don't have enough evidence to reject the null hypothesis at 5% of significance.

3 0
3 years ago
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