3.16 Grades on a standardized test are known to have a mean of 1000 for students

Question

3.16 Grades on a standardized test are known to have a mean of 1000 for students in the United States. The test is administered to 453 randomly selected students in Florida; in this sample, the mean is 1013, and the stan- dard deviation (s) is 108. a. Construct a 95% confidence interval for the average test score for Florida students. b. Is there statistically significant evidence that Florida students perform differently than other students in the United States? c. Another 503 students are selected at random from Florida. They are given a 3-hour preparation course before the test is adminis tered. Their average test score is 1019, with a standard deviation of 95 i. Construct a 95% confidence interval for the change in average test score associated with the prep course. ii. Is there statistically significant evidence that the prep course helped?

Explanation / Answer

Here we have to test the hypothesis that,

H0 : mu = 1000 Vs H1 : mu not= 1000

where mu is population mean for students in the United states.

Assume alpha =level of significance = 0.05

Here sample size is too large so we use one sample z-test.

And also we have to find 95% confidence interval for population mean (mu).

95% confidence interval for mu is,

Xbar - E < mu < Xbar + E

Xbar is sample mean.

where E is margin of error.

E = (Zc * s) / sqrt(n)

where Zc is critical value of normal distribution.

s is standard deviation.

n is sample size.

Given that,

sample size (n) = 453

Sample mean (Xbar) = 1013

Standard deviation (s) = 108

a) We can do first two parts in TI-83 calculator.

steps :

STAT --> TESTS --> 1 : Z-Test --> ENTER --> High light on Stats --> ENTER --> Input all the values --> select alternative "not= mu0" --> ENTER --> Calculate--> ENTER

Test statistic = 2.56

P-value = 0.0104

P-value < alpha

Reject H0 at 5% level of significance.

Conclusion : There is sufficient evidence to say that the population mean for students in the United state will be differ than 1000.

Now we have to find 95% confidence interval for population mean (mu).

b) Steps in TI-83 calculator :

STAT --> TESTS --> 7:ZInterval --> ENTER --> Highlight on Stats --> ENTER --> Input all the values --> Calculate --> ENTER

95% confidence interval for mu is (1003.1, 1022.9).

We are 95% confident that the population mean lies between 1003.1 and 1022.9

c) Now similar procedure we have to do for the different data.

n = 503

Xbar= 1019

s = 95

One sample Z Test :

Test statistic = 4.49

P-value = 7.279797E-6 = 0.0000

P-value < alpha

Reject H0 at 5% level of significance.

Conclusion : There is sufficient evidence to say that the population mean for students in the United state will be differ than 1000.

95% Confidence interval :

95% confidence interval for mu is (1010.7, 1027.3).

We are 95% confident that the population mean lies between 1003.1 and 1022.9

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