Statistics - Baseball Team Scores

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See the attachment please.

1. Develop a 90% confidence interval and a 95% confidence interval for the population mean. Did you use a z statistic or t statistics? Why? Compare the size and meaning of the two confidence intervals.

We use the column-E: R. By using Excel, we get the following descriptive statistics
Column-E

Mean 281.1667
Standard Error 4.803276
Median 282.5

Mode 285
Standard Deviation 26.30862
Sample Variance 692.1437
Kurtosis -0.29514
Skewness 0.128215
Range 112
Minimum 222
Maximum 334
Sum 8435
Count 30

From above table, we know that the sample size n=30, sample mean =281.1667 and sample standard deviation s=26.30862. Use a website http://duke.usask.ca/~rbaker/Tables.html , we get .
So, a 90% confidence interval and a 95% confidence interval for the population mean are

and

i.e.,

and

We use t-statistics since we don't know the standard deviation of population. A 95% Confidence interval has a larger length than a 90% CI does. It tells us that the population mean lies in with probability 90%, and it lies in with 95% probability.

2. Test a hypothesis about a population mean. Take a random sample and perform a t-test against a population mean. Explain what assumptions you made.

We use the column-E: R. By using Excel, we get the following descriptive statistics
Column-E

Mean 281.1667
Standard Error 4.803276
Median 282.5

Mode 285
Standard Deviation 26.30862
Sample Variance 692.1437
Kurtosis -0.29514
Skewness 0.128215
Range 112
Minimum 222
Maximum 334
Sum 8435
Count 30

WE try to test the following hypotheses. Denote the population mean by .

From above table, we know that the sample size n=30, sample mean =281.1667 and sample standard deviation s=26.30862. By question 1), we know that a 95% confidence interval is . Now 270 is NOT in
So, with 5% significance level, we should reject the null hypothesis We conclude that

3. Test a hypothesis comparing two population means. Collect two independent samples. Perform a t-test using two independent samples.

We use the column-U and V: HBB-IBB. Use Excel, we get the following

t-Test: Two-Sample Assuming Equal Variances

Variable HBB Variable IBB
Mean 22.1 15.5
Variance 36.57586 26.32759
Observations 30 30
Pooled Variance 31.45172
Hypothesized Mean Difference 0
df 58
t Stat 4.557927
P(T<=t) one-tail 1.36E-05
t Critical one-tail 1.671553
P(T<=t) two-tail 2.71E-05
t Critical two-tail 2.001717

From the above table, we know that p-value=P(T<=t) two-tail=2.71E-05<0.05. So, with 5% significance level, we should reject the null hypothesis: No difference between their means. We conclude that there is a significant difference between their means.

4. Test a hypothesis comparing two population means using paired (matched) observations. Collect two dependent samples. Perform a t test for paired observations.

We use the column-U and V: HBB-IBB. Now we assume that two samples are dependent. Use Excel, we get the following

t-Test: Paired Two Sample for Means

Variable HBB Variable IBB
Mean 22.1 15.5
Variance 36.57586 26.32759
Observations 30 30
Pearson Correlation -0.08723
Hypothesized Mean Difference 0
df 29
t Stat 4.373603
P(T<=t) one-tail 7.2E-05
t Critical one-tail 1.699127
P(T<=t) two-tail 0.000144
t Critical two-tail 2.04523

From the above table, we know that ...