Question

During the past year, on the average, 289 books were checked out daily from the public library in Stucco, Wyoming. This year, the city fathers of Stucco have attempted to increase circulation through various promotions such as bookmobiles, youth programs, and visits to local organizations. In order to evaluate their efforts, they have taken a random sample of 50 days of the current year and calculated the mean number of books checked out daily --288-- and the standard deviation--23. Based on this data, it is likely (95% confidence level) that the daily circulation of books has not increased? Please show math step by step. how you would put it in a TI-84 calculator to arrive at your answer

Answer 1

Solution:-

95% confidence interval for the mean is C.I = ( 281.46, 294.54).

C.I = 288 + 2.01*3.2527

C.I = 288 + 6.53793

C.I = ( 281.46, 294.54)

From the above test we have sufficient evidence it is likely that the daily circulation of books has not increased, because the confidence interval contains values less than 289.

Hypothesis test

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u < 289
Alternative hypothesis: u > 289

Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the sample mean is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. The test method is a one-sample t-test.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = s / sqrt(n)

S.E = 3.2527
DF = n - 1

D.F = 49
t = (x - u) / SE

t = - 0.3074

where s is the standard deviation of the sample, x is the sample mean, u is the hypothesized population mean, and n is the sample size.

The observed sample mean produced a t statistic test statistic of -0.3074

Thus the P-value in this analysis is 0.62.

Interpret results. Since the P-value (0.62) is greater than the significance level (0.05), we cannot reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that the daily circulation of books has not increased.