Question

To test Upper H 0 : sigma equals 2.4 versus Upper H 1 : sigma greater than 2.4​, a random sample of size n equals 17 is obtained from a population that is known to be normally distributed. Complete parts​ (a) through​ (d). ​(a) If the sample standard deviation is determined to be s equals 2.3​, compute the test statistic. chi Subscript 0 Superscript 2equals nothing ​(Round to three decimal places as​ needed.) ​(b) If the researcher decides to test this hypothesis at the alpha equals 0.01 level of​ significance, determine the critical value. chi Subscript 0.01 Superscript 2equals nothing ​(Round to three decimal places as​ needed.) ​(c) Draw a​ chi-square distribution and depict the critical region. 32 14.69 A graph has a horizontal axis with two labeled coordinates at 14.69 and 32 and an unlabeled vertical axis. The graph contains a chi-square distribution curve that rises from left to right at an increasing and then decreasing rate to a peak about one quarter of the way from the left edge of the graph and then falls from left to right at an increasing and then decreasing rate to the right edge of the graph. There are two vertical lines that extend from the horizontal axis to the curve. The leftmost line is at 14.69 and is close to the peak. The rightmost line is at 32 and is to the right of the peak. The region under the curve to the right of the rightmost line is shaded. 5.812 32.000 A graph has a horizontal axis with two labeled coordinates at 5.812 and 32.000 and an unlabeled vertical axis. The graph contains a chi-square distribution curve that rises from left to right at an increasing and then decreasing rate to a peak about one quarter of the way from the left edge of the graph and then falls from left to right at an increasing and then decreasing rate to the right edge of the graph. There are two vertical lines that extend from the horizontal axis to the curve. The leftmost line is at 5.812 and is to the left of the peak. The rightmost line is at 32.000 and is to the right of the peak. The regions under the curve between the vertical axis and the leftmost line and to the right of the rightmost line are shaded. 32 14.69 A graph has a horizontal axis with two labeled coordinates at 14.69 and 32 and an unlabeled vertical axis. The graph contains a chi-square distribution curve that rises from left to right at an increasing and then decreasing rate to a peak about one quarter of the way from the left edge of the graph and then falls from left to right at an increasing and then decreasing rate to the right edge of the graph. There are two vertical lines that extend from the horizontal axis to the curve. The leftmost line is at 14.69 and is close to the peak. The rightmost line is at 32 and is to the right of the peak. The region under the curve between the two vertical lines is shaded. 5.812 32.000 A graph has a horizontal axis with two labeled coordinates at 5.812 and 32.000 and an unlabeled vertical axis. The graph contains a chi-square distribution curve that rises from left to right at an increasing and then decreasing rate to a peak about one quarter of the way from the left edge of the graph and then falls from left to right at an increasing and then decreasing rate to the right edge of the graph. There are two vertical lines that extend from the horizontal axis to the curve. The leftmost line is at 5.812 and is to the left of the peak. The rightmost line is at 32.000 and is to the right of the peak. The region under the curve between the vertical axis and the leftmost line is shaded. ​(d) Will the researcher reject the null​ hypothesis? Reject Upper H 0 because chi Subscript 0 Superscript 2greater thanchi Subscript 0.01 Superscript 2. Do not reject Upper H 0 because chi Subscript 0 Superscript 2greater thanchi Subscript 0.01 Superscript 2

Answer 1

Given that a sample of size, n, 17 is drawn from the normal population.

To test for the hypothesis,

: = 2.4

versus

: > 2.4.

(a) Given that sample standard deviation, s =2.3.

To find: the test statistic

,

where is the value of under .

Then, we get

(16*2.3^2)/2.4^2 = 14.695.

(b) At = 0.01 level of significance, the critical value of Chi-square given by

= 32.

Note that the Chi-square critical values table gives the area to the right or the greater than values.

(c)The Chi-square curve

(d)Decision:

Since < , we do not reject the and conclude that = 2.4.